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Mathematics and Statistics · Class 11 Commerce

Ch 15Permutations and Combinations — Class 11 Mathematics and Statistics, concept-first.

This chapter of the Maharashtra Std XI (FYJC) commerce Mathematics and Statistics course is about counting without listing — finding how many arrangements or selections are possible when writing every one out by hand would be hopeless.

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Concepts

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Key concepts

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Permutations

Imagine you have three different books — a Physics book, a Chemistry book, and a Maths book — and you want to place them on a shelf. How many different ways can you arrange them?

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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.

1

The Fundamental Principle of Counting

This chapter of the Maharashtra Std XI (FYJC) commerce Mathematics and Statistics course is about counting without listing — finding how many arrangements or selections are possible when writing every…

2

Factorial Notation

Counting arrangements produces products such as so often that a short symbol is used for them.

3

Permutations: Arrangements in Order

A permutation is an arrangement of objects in a definite order. Order matters: the arrangement is counted as different from .

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Permutations with Identical Objects and with Restrictions

Some of the objects are identical. When a set of objects contains groups of alike objects, swapping two identical objects does not create a new arrangement, so the plain over-counts.

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Circular Permutations

When objects are arranged around a circle (people around a round table, beads spaced on a ring), there is no fixed "first" position — rotating everyone one seat to the left gives the same circular arr…

6

Combinations: Selections Without Order

A combination is a selection of objects where order does not matter — only which objects are chosen, not the order in which they are picked. Choosing is the same combination as .

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Properties of Combinations

The combination symbol obeys several identities that shorten calculations and are frequently examined.

Exercises

+Show 8 questions8 questions
  1. Q11How many arrangements can be made of the letters of the word $\textbf{BALLOON}$?Free
  2. Q12There are $12$ points in a plane, no three of which are collinear. (i) How many straight lines can be drawn through them? (ii) How many tria…Free
  3. Q13A cricket team of $11$ players is to be chosen from $15$ players, of whom $5$ are bowlers. In how many ways can the team be chosen so that i…Free
  4. Q14In a meeting, every participant shook hands with every other participant exactly once. If there were $66$ handshakes in all, how many partic…Preview
  5. Q15How many diagonals does a polygon with $10$ sides (a decagon) have?Preview
  6. Q16From $7$ consonants and $4$ vowels, how many words (arrangements) can be formed each consisting of $3$ consonants and $2$ vowels? (The "word…Preview
  7. Q17In how many ways can $8$ different beads be arranged to form a necklace?Preview
  8. Q18A person has $6$ friends. In how many ways can he invite one or more of them to a dinner party?Preview

More questions

+Show 10 questions10 questions
  1. Example 1How many $3$-digit numbers can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated?Free
  2. Example 2A student can travel from city A to city B by $3$ bus services or $2$ train services, and from B to C by $4$ bus services or $1$ flight. In…Free
  3. Example 3Evaluate $\dfrac{9!}{6!\,3!}$.Free
  4. Example 4Find $n$ if $^{n}P_{3} = 60$.Preview
  5. Example 5In how many ways can the letters of the word $\textbf{DELHI}$ be arranged? In how many of these do the two vowels come together?Preview
  6. Example 6How many distinct arrangements can be made of the letters of the word $\textbf{STATISTICS}$?Preview
  7. Example 7How many $4$-digit PIN codes can be formed from the digits $0$ to $9$ if repetition of digits is allowed?Preview
  8. Example 8In how many ways can $6$ people be seated around a circular table? In how many of these arrangements do two particular people, X and Y, sit…Preview
  9. Example 9A committee of $3$ members is to be selected from $8$ people. In how many ways can this be done?Preview
  10. Example 10Find $r$ if $^{15}C_{r} = {}^{15}C_{r+3}$.Preview