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Mathematics · Class 11 Science

Ch 12Permutations and Combination — Class 11 Mathematics, concept-first.

Counting is one of the very first mathematical skills we learn, and as the objects to be counted grow large, simply counting them one by one stops being practical.

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Key concepts

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Fundamental Principle of Counting

The Fundamental Principle of Counting is the basic toolkit for counting outcomes without listing them one by one, and it has two complementary halves.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

3.1

Introduction

Counting is one of the very first mathematical skills we learn, and as the objects to be counted grow large, simply counting them one by one stops being practical.

+EXERCISE 3.114 questions
  1. Q1A teacher wants to select the class monitor in a class of 30 boys and 20 girls. In how many ways can the monitor be selected if the monitor…Free
  2. Q2A Signal is generated from 2 flags by putting one flag above the other. If 4 flags of different colours are available, how many different si…Free
  3. Q3How many two letter words can be formed using letters from the word SPACE, when repetition of letters (i) is allowed, (ii) is not allowed?Free
  4. Q4How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits (i) are allowed, (ii) are not allowed?Preview
  5. Q5How many three-digit numbers can be formed using the digits 2, 3,4,5,6 if digits can be repeated?Preview
  6. Q6A letter lock contains 3 rings and each ring containing 5 different letters. Determine the maximum number of false trials that can be made b…Preview
  7. Q7In a test, 5 questions are of the form 'state, true or false'. No student has got all answers correct. Also, the answer of every student is…Preview
  8. Q8How many numbers between 100 and 1000 have 4 in the units place?Preview
  9. Q9How many numbers between 100 and 1000 have the digit 7 exactly once?Preview
  10. Q10How many four digit numbers will not exceed 7432 if they are formed using the digits 2,3,4,7 without repetition?Preview
  11. Q11If numbers are formed using digits 2, 3, 4, 5, 6 without repetition, how many of them will exceed 400?Preview
  12. Q12How many numbers formed with the digits 0, 1, 2, 5, 7, 8 will fall between 13 and 1000 if digits can be repeated?Preview
  13. Q13A school has three gates and four staircases from the first floor to the second floor. How many ways does a student have to go from outside…Preview
  14. Q14How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?Preview
3.2

Fundamental principles of counting

This section lays the groundwork for every counting technique in the chapter: the two Fundamental Principles of Counting (Addition and Multiplication), introduced through the visual device of a tree d…

+EXERCISE 3.241 questions
  1. Q15Evaluate: 8!Free
  2. Q16Evaluate: 10!Free
  3. Q17Evaluate: 10! - 6!Free
  4. Q18Evaluate: (10 - 6)!Preview
  5. Q19Compute: 12!/6!Preview
  6. Q20Compute: ${}^{12}C_6$Preview
  7. Q21Compute: (3 × 2)!Preview
  8. Q22Compute: 3! × 2!Preview
  9. Q23Compute: 9!/(3! 6!)Preview
  10. Q24Compute: (6! - 4!)/4!Preview
  11. Q25Compute: 8!/(6! - 4!)Preview
  12. Q26Compute: 8!/(6-4)!Preview
  13. Q27Write in terms of factorials: 5 × 6 × 7 × 8 × 9 × 10Preview
  14. Q28Write in terms of factorials: 3 × 6 × 9 × 12 × 15Preview
  15. Q29Write in terms of factorials: 6 × 7 × 8 × 9Preview
  16. Q30Write in terms of factorials: 5 × 10 × 15 × 20Preview
  17. Q31Evaluate n!/(r!(n-r)!) for n = 8, r = 6Preview
  18. Q32Evaluate n!/(r!(n-r)!) for n = 12, r = 12Preview
  19. Q33Evaluate n!/(r!(n-r)!) for n = 15, r = 10Preview
  20. Q34Evaluate n!/(r!(n-r)!) for n = 15, r = 8Preview
  21. Q35Find n, if: [equation involving n, 8!, 6!, 4! — a stacked-fraction expression whose exact layout could not be confidently reconstructed from…Preview
  22. Q36Find n, if: [equation involving n, 8!, 6!, 3! — a stacked-fraction expression whose exact layout could not be confidently reconstructed from…Preview
  23. Q37Find n, if: [equation involving n!, 4!, 5! — a stacked-fraction expression whose exact layout could not be confidently reconstructed from th…Preview
  24. Q38Find n, if: $(n+1)! = 42\times(n-1)!$Preview
  25. Q39Find n, if: $(n+3)! = 110\times(n+1)!$Preview
  26. Q40Find n, if: $\dfrac{(17-n)!}{(14-n)!} = 5!$Preview
  27. Q41Find n, if: $\dfrac{(15-n)!}{(13-n)!} = 12$Preview
  28. Q42Find n, if: ${}^nC_3 : {}^nC_5 = 5 : 3$Preview
  29. Q43Find n, if: ${}^nC_3 : {}^nC_7 = 1 : 6$Preview
  30. Q44Find n, if: ${}^{2n}C_7 : {}^nC_4 = 24 : 1$Preview
  31. Q45Simplify: [algebraic factorial expression, item (i) — a nested stacked-fraction layout whose exact bracket placement could not be confidentl…Preview
  32. Q46Simplify: [algebraic factorial expression, item (ii) — a nested stacked-fraction layout whose exact bracket placement could not be confident…Preview
  33. Q47Simplify: [algebraic factorial expression, item (iii) — a nested stacked-fraction layout whose exact bracket placement could not be confiden…Preview
  34. Q48Simplify: [algebraic factorial expression, item (iv) — a nested stacked-fraction layout whose exact bracket placement could not be confident…Preview
  35. Q49Simplify: [algebraic factorial expression, item (v) — a nested stacked-fraction layout whose exact bracket placement could not be confidentl…Preview
  36. Q50Simplify: [algebraic factorial expression, item (vi) — a nested stacked-fraction layout whose exact bracket placement could not be confident…Preview
  37. Q51Simplify: [algebraic factorial expression, item (vii) — a nested stacked-fraction layout whose exact bracket placement could not be confiden…Preview
  38. Q52Simplify: [algebraic factorial expression, item (viii) — a nested stacked-fraction layout whose exact bracket placement could not be confide…Preview
  39. Q53Show that $\dfrac{n!}{r!(n-r)!} + \dfrac{n!}{(r-1)!(n-r+1)!} = \dfrac{(n+1)!}{r!(n-r+1)!}$Preview
  40. Q54Show that $\dfrac{9!}{3!6!} + \dfrac{9!}{4!5!} = \dfrac{10!}{4!6!}$Preview
  41. Q55Show that $\dfrac{(2n)!}{n!} = 2^n\,\{1\cdot3\cdot5\cdots(2n-1)\}$Preview
3.2.1

Addition Principle

This sub-section states the Addition Principle formally: suppose one operation can be done in ways and another operation can be done in ways, with no way common to the two operations.

3.2.2

Multiplication principle

This sub-section states the Multiplication Principle formally: if one operation can be carried out in ways, followed by a second operation that can be carried out in ways, and the two operations are i…

3.3

Invariance Principle

This short section states the Invariance Principle: the result of counting the objects in a set does not depend on the order in which the objects are counted, nor on the particular method used to coun…

+EXERCISE 3.333 questions
  1. Q56Find n, if ${}^nP_6 : {}^nP_3 = 120:1$Free
  2. Q57Find m and n, if $(m+n)P_2 = 56$ and $(m-n)P_2 = 12$Free
  3. Q58Find r, if ${}^{12}P_{r-2} : {}^{11}P_{r-1} = 3:14$Free
  4. Q59Show that $(n+1)\,({}^nP_r) = (n-r+1)\,[{}^{(n+1)}P_r]$Preview
  5. Q60How many 4 letter words can be formed using letters in the word MADHURI if letters can be repeated?Preview
  6. Q61How many 4 letter words can be formed using letters in the word MADHURI if letters cannot be repeated?Preview
  7. Q62Determine the number of arrangements of letters of the word ALGORITHM if vowels are always together.Preview
  8. Q63Determine the number of arrangements of letters of the word ALGORITHM if no two vowels are together.Preview
  9. Q64Determine the number of arrangements of letters of the word ALGORITHM if consonants are at even positions.Preview
  10. Q65Determine the number of arrangements of letters of the word ALGORITHM if O is the first and T is the last letter.Preview
  11. Q66In a group photograph, 6 teachers are in the first row and 18 students are in the second row. There are 12 boys and 6 girls among the studen…Preview
  12. Q67Find the number of ways so that letters of the word HISTORY can be arranged as: Y and T are togetherPreview
  13. Q68Find the number of ways so that letters of the word HISTORY can be arranged as: Y is next to TPreview
  14. Q69Find the number of ways so that letters of the word HISTORY can be arranged as: there is no restrictionPreview
  15. Q70Find the number of ways so that letters of the word HISTORY can be arranged as: begin and end with vowelPreview
  16. Q71Find the number of ways so that letters of the word HISTORY can be arranged as: end in STPreview
  17. Q72Find the number of ways so that letters of the word HISTORY can be arranged as: begin with S and end with TPreview
  18. Q73Find the number of arrangements of the letters in the word SOLAPUR so that consonants and vowels are placed alternately.Preview
  19. Q74Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 8 if digits can be repeatedPreview
  20. Q75Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 8 if digits cannot be repeatedPreview
  21. Q76How many numbers can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition so that resulting numbers are between 100 and 1000?Preview
  22. Q77Find the number of 6-digit numbers using the digits 3,4,5,6,7,8 without repetition. How many of these numbers are (a) divisible by 5, (b) no…Preview
  23. Q78A code word is formed by two different English letters followed by two non-zero distinct digits. Find the number of such code words. Also, f…Preview
  24. Q79Find the number of ways in which 5 letters can be posted in 3 post boxes if any number of letters can be posted in a post box.Preview
  25. Q80Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object always occurs.Preview
  26. Q81Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object never occurs.Preview
  27. Q82In how many ways can 5 different books be arranged on a shelf if there are no restrictions?Preview
  28. Q83In how many ways can 5 different books be arranged on a shelf if 2 books are always together?Preview
  29. Q84In how many ways can 5 different books be arranged on a shelf if 2 books are never together?Preview
  30. Q853 boys and 3 girls are to sit in a row. How many ways can this be done if there are no restrictions?Preview
  31. Q863 boys and 3 girls are to sit in a row. How many ways can this be done if there is a girl at each end?Preview
  32. Q873 boys and 3 girls are to sit in a row. How many ways can this be done if boys and girls are at alternate places?Preview
  33. Q883 boys and 3 girls are to sit in a row. How many ways can this be done if all boys sit together?Preview
3.4

Factorial Notation

This section introduces factorial notation, the essential shorthand that underlies every formula developed later in the chapter for permutations and combinations.

+EXERCISE 3.422 questions
  1. Q89Find the number of permutations of letters in the word DIVYA.Free
  2. Q90Find the number of permutations of letters in the word SHANTARAM.Free
  3. Q91Find the number of permutations of letters in the word REPRESENT.Free
  4. Q92Find the number of permutations of letters in the word COMBINE.Preview
  5. Q93Find the number of permutations of letters in the word BALBHARATI.Preview
  6. Q94You have 2 identical books on English, 3 identical books on Hindi, and 4 identical books on Mathematics. Find the number of distinct ways of…Preview
  7. Q95A coin is tossed 8 times. In how many ways can we obtain 4 heads and 4 tails?Preview
  8. Q96A coin is tossed 8 times. In how many ways can we obtain at least 6 heads?Preview
  9. Q97A bag has 5 red, 4 blue, and 4 green marbles. If all are drawn one by one and their colours are recorded, how many different arrangements ca…Preview
  10. Q98Find the number of ways of arranging letters of the word MATHEMATICAL.Preview
  11. Q99Find the number of ways of arranging letters of the word MATHEMATICAL so that all vowels are together.Preview
  12. Q100Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have (a) letters R and H never…Preview
  13. Q101How many different words are formed if the letters R is used thrice and letters S and T are used twice each?Preview
  14. Q102Find the number of arrangements of letters in the word MUMBAI so that the letter B is always next to A.Preview
  15. Q103Find the number of arrangements of letters in the word CONSTITUTION that begin and end with N.Preview
  16. Q104Find the number of different ways of arranging letters in the word ARRANGE. How many of these arrangement do not have the two R's and two A'…Preview
  17. Q105How many distinct 5 digit numbers can be formed using the digits 3, 2, 3, 2, 4, 5.Preview
  18. Q106Find the number of distinct numbers formed using the digits 3, 4, 5, 6, 7, 8, 9, so that odd positions are occupied by odd digits.Preview
  19. Q107How many different 6-digit numbers can be formed using digits in the number 659942? How many of them are divisible by 4?Preview
  20. Q108Find the number of distinct words formed from letters in the word INDIAN. How many of them have the two N's together?Preview
  21. Q109Find the number of different ways of arranging letters in the word PLATOON if the two O's are never together.Preview
  22. Q110Find the number of different ways of arranging letters in the word PLATOON if consonants and vowels occupy alternate positions.Preview
3.5

Permutations: (When all objects are distinct)

This section builds up to the formal idea of a permutation using small, concrete seating examples before the general theorem is proved in the next sub-section.

+EXERCISE 3.513 questions
  1. Q111In how many different ways can 8 friends sit around a table?Free
  2. Q112A party has 20 participants. Find the number of distinct ways for the host to sit with them around a circular table.Free
  3. Q113A party has 20 participants. How many of these ways have two specified persons on either side of the host?Free
  4. Q114Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangements where two specified delegates a…Preview
  5. Q115Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangements where two specified delegates a…Preview
  6. Q116Find the number of ways for 15 people to sit around the table so that no two arrangements have the same neighbours.Preview
  7. Q117A committee of 10 members sits around a table. Find the number of arrangements that have the president and the vice president together.Preview
  8. Q118Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated between the two women.Preview
  9. Q119Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated between two men.Preview
  10. Q120Eight men and six women sit around a table. How many of sitting arrangements will have no two women together?Preview
  11. Q121Find the number of seating arrangements for 3 men and 3 women to sit around a table so that exactly two women are together.Preview
  12. Q122Four objects in a set of ten objects are alike. Find the number of ways of arranging them in a circular order.Preview
  13. Q123Fifteen persons sit around a table. Find the number of arrangements that have two specified persons not sitting side by side.Preview
3.5.1

Permutations when all objects are distinct [r ≤ n]

This sub-section proves Theorem 1: the number of permutations of distinct objects taken at a time (with ), without repetitions, is .

3.5.2

Permutations when repetitions are allowed

This sub-section proves Theorem 2: when repetitions ARE allowed, the number of arrangements of distinct objects taken at a time is .

3.5.3

Permutations when some objects are identical

This sub-section extends the permutation formula to the case where some of the objects being arranged are actually IDENTICAL to each other — a situation that arises naturally when arranging the letter…

3.5.4

Circular permutation

This sub-section develops the formula for arranging objects around a CIRCLE, rather than in a straight row — a genuinely different counting problem, because a circular arrangement has no fixed startin…

3.6

Combinations

This section introduces the COMBINATION — a fundamentally different counting question from the permutation, since a combination is a SELECTION of objects where the ORDER of selection is irrelevant, in…

+EXERCISE 3.642 questions
  1. Q124Find the value of ${}^{15}C_4$Free
  2. Q125Find the value of ${}^{80}C_2$Free
  3. Q126Find the value of ${}^{15}C_4 + {}^{15}C_5$Free
  4. Q127Find the value of ${}^{20}C_{16} - {}^{19}C_{16}$Preview
  5. Q128Find n if ${}^6P_2 = n\cdot{}^6C_2$Preview
  6. Q129Find n if ${}^{2n}C_3 : {}^nC_2 = 52 : 3$Preview
  7. Q130Find n if ${}^nC_{n-3} = 84$Preview
  8. Q131Find r if ${}^{14}C_{2r} : {}^{10}C_{2r-4} = 143:10$Preview
  9. Q132Find n and r if ${}^nP_r = 720$ and ${}^nC_{n-r} = 120$Preview
  10. Q133Find n and r if ${}^nC_{r-1} : {}^nC_r : {}^nC_{r+1} = 20:35:42$Preview
  11. Q134If ${}^nP_r = 1814400$ and ${}^nC_r = 45$, find ${}^{n+4}C_{r+3}$Preview
  12. Q135If ${}^nC_{r-1} = 6435$, ${}^nC_r = 5005$, ${}^nC_{r+1} = 3003$, find ${}^rC_5$Preview
  13. Q136Find the number of ways of drawing 9 balls from a bag that has 6 red balls, 8 green balls, and 7 blue balls so that 3 balls of every colour…Preview
  14. Q137Find the number of ways of selecting a team of 3 boys and 2 girls from 6 boys and 4 girls.Preview
  15. Q138After a meeting, every participant shakes hands with every other participants. If the number of handshakes is 66, find the number of partici…Preview
  16. Q139If 20 points are marked on a circle, how many chords can be drawn?Preview
  17. Q140Find the number of diagonals of an n-sided polygon. In particular, find the number of diagonals when (a) n = 10 (b) n = 15 (c) n = 12 (d) n…Preview
  18. Q141There are 20 straight lines in a plane so that no two lines are parallel and no three lines are concurrent. Determine the number of points o…Preview
  19. Q142Ten points are plotted on a plane. Find the number of straight lines obtained by joining these points if no three points are collinear.Preview
  20. Q143Ten points are plotted on a plane. Find the number of straight lines obtained by joining these points if four points are collinear.Preview
  21. Q144Find the number of triangles formed by joining 12 points if no three points are collinear.Preview
  22. Q145Find the number of triangles formed by joining 12 points if four points are collinear.Preview
  23. Q146A word has 8 consonants and 3 vowels. How many distinct words can be formed if 4 consonants and 2 vowels are chosen?Preview
  24. Q147Find n if ${}^nC_8 = {}^nC_{12}$Preview
  25. Q148Find n if ${}^{23}C_{3n} = {}^{23}C_{2n+3}$Preview
  26. Q149Find n if ${}^{21}C_{6n} = {}^{21}C_{n^2+5}$Preview
  27. Q150Find n if ${}^{2n}C_{r-1} = {}^{2n}C_{r+1}$Preview
  28. Q151Find n if ${}^nC_{n-2} = 15$Preview
  29. Q152Find x if ${}^nP_r = x\cdot{}^nC_r$Preview
  30. Q153Find r if ${}^{11}C_4+ {}^{11}C_5 + {}^{12}C_6 + {}^{13}C_7 = {}^{14}C_r$Preview
  31. Q154Find the value of $\displaystyle\sum_{r=1}^{4} {}^{(21-r)}C_{4}$Preview
  32. Q155Find the difference between the greatest values of ${}^{14}C_r$ and ${}^{12}C_r$Preview
  33. Q156Find the difference between the greatest values of ${}^{13}C_r$ and ${}^{8}C_r$Preview
  34. Q157Find the difference between the greatest values of ${}^{15}C_r$ and ${}^{11}C_r$Preview
  35. Q158In how many ways can a boy invite his 5 friends to a party so that at least three join the party?Preview
  36. Q159A group consists of 9 men and 6 women. A team of 6 is to be selected. How many of possible selections will have at least 3 women?Preview
  37. Q160A committee of 10 persons is to be formed from a group of 10 women and 8 men. How many possible committees will have at least 5 women?Preview
  38. Q161A committee of 10 persons is to be formed from a group of 10 women and 8 men. How many possible committees will have men in majority?Preview
  39. Q162A question paper has two sections. section I has 5 questions and section II has 6 questions. A student must answer at least two question fro…Preview
  40. Q163There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicket…Preview
  41. Q164Five students are selected from 11. How many ways can these students be selected if two specified students are selected?Preview
  42. Q165Five students are selected from 11. How many ways can these students be selected if two specified students are not selected?Preview
3.6.1

Properties of combinations

This sub-section lists nine standard Properties of Combinations and then works through seven varied solved examples that make heavy use of them.

More questions

32 Q
+Show 10 questions10 questions
  1. Q166A college offers 5 courses in the morning and 3 in the evening. The number of ways a student can select exactly one course, either in the mo…Free
  2. Q167A college has 7 courses in the morning and 3 in the evening. The possible number of choices with the student if he wants to study one course…Free
  3. Q168In how many ways can 8 Indians and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit togeth…Free
  4. Q169In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together? A) 9×8! B) 8×8! C) 9×9! D)…Preview
  5. Q170In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate. A) 12 B) 288 C) 144 D) 256Preview
  6. Q171Find the number of triangles which can be formed by joining the angular points of a polygon of 8 sides as vertices. A) 16 B) 56 C) 24 D) 8Preview
  7. Q172A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 8 from part A and 5 from part B, In how ma…Preview
  8. Q173There are 10 persons among whom two are brothers. The total number of ways in which these persons can be seated around a round table so that…Preview
  9. Q174The number of arrangements of the letters of the word BANANA in which two N's do not appear adjacently. A) 80 B) 60 C) 40 D) 100Preview
  10. Q175The number of ways in which 5 male and 2 female members of a committee can be seated around a round table so that the two females are not se…Preview
+Show 22 questions22 questions
  1. Q176Find the value of r if ${}^{56}P_{r+6} : {}^{54}P_{r+3}= 30800:1$Free
  2. Q177How many words can be formed by writing letters in the word CROWN in different order?Free
  3. Q178Find the number of words that can be formed by using all the letters in the word REMAIN.Free
  4. Q179If these words (formed from all letters of REMAIN) are written in dictionary order, what will be the 40th word?Preview
  5. Q180Capital English alphabet has 11 symmetric letters that appear same when looked at in a mirror. These letters are A, H, I, M, O, T, U, V, W,…Preview
  6. Q181How many numbers formed using the digits 3,2,0,4,3,2,3 exceed one million?Preview
  7. Q182Ten students are to be selected for a project from a class of 30 students. There are 4 students who want to be together either in the projec…Preview
  8. Q183A student finds 7 books of his interest, but can borrow only three books. He wants to borrow Chemistry part II book only if Chemistry Part I…Preview
  9. Q18430 objects are to be divided in three groups containing 7,10,13 objects. Find the number of distinct ways for doing so.Preview
  10. Q185A student passes an examination if he secures a minimum in each of the 7 subjects. Find the number of ways a student can fail.Preview
  11. Q186Nine friends decide to go for a picnic in two groups. One group decides to go by car and the other group decides to go by train. Find the nu…Preview
  12. Q187A hall has 12 lamps and every lamp can be switched on independently. Find the number of ways of illuminating the hall.Preview
  13. Q188How many quadratic equations can be formed using numbers from 0,2,4,5 as coefficients if a coefficient can be repeated in an equation.Preview
  14. Q189How many six-digit telephone numbers can be formed if the first two digits are 45 and no digit can appear more than once?Preview
  15. Q190A question paper has 6 questions. How many ways does a student have to answer if he wants to solve at least one question?Preview
  16. Q191Find the number of ways of dividing 20 objects in three groups of sizes 8,7,and 5.Preview
  17. Q192There are 4 doctors and 8 lawyers in a panel. Find the number of ways for selecting a team of 6 if at least one doctor must be in the team.Preview
  18. Q193Four parallel lines intersect another set of five parallel lines. Find the number of distinct parallelograms formed.Preview
  19. Q194There are 12 distinct points A,B,C,.....,L, in order, on a circle. Lines are drawn passing through each pair of points. How many lines are t…Preview
  20. Q195There are 12 distinct points A,B,C,.....,L, in order, on a circle. How many lines pass through D?Preview
  21. Q196There are 12 distinct points A,B,C,.....,L, in order, on a circle. How many triangles are determined by lines.Preview
  22. Q197There are 12 distinct points A,B,C,.....,L, in order, on a circle. How many triangles have one vertex C?Preview