Mathematics · Class 11 Science
Ch 6Permutations and Combinations — Class 11 Mathematics, concept-first.
Many counting problems in real life boil down to a sequence of independent choices, each with a known number of possible outcomes. The Fundamental Principle of Counting (FPC) — also called the multiplication principle — gives the basic rule for combining such choices.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
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?
Most relevant Q&A
- How many different signals can be made by hoisting 2 flags at a time, one above the other on a vertical pole, chosen from 4 flags of differe…Free
- How many 4-letter words (not necessarily meaningful) can be formed using the letters of the word \"ORANGE\" (all 6 letters distinct), withou…Preview
- Evaluate $^{7}P_{3}$.Free
- How many 5-letter words can be formed using the letters of the word \"NUMBER\" (6 distinct letters), without repeating any letter?Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Fundamental Principle of Counting
Many counting problems in real life boil down to a sequence of independent choices, each with a known number of possible outcomes.
Factorial Notation
Many of the counting formulas developed later in this chapter involve repeated products such as . Because this pattern occurs so often, it is given its own compact notation.
Permutations — Definition and Formula Derivation
A permutation is an arrangement of a given number of objects taken from a larger collection, where the order in which the objects are placed matters — swapping two objects' positions produces a differ…
Permutations with Repetition and Restrictions
The formula from Section 3 assumes every object is distinct and none is repeated in the arrangement. Three common variations extend this basic idea.
Combinations — Definition and Formula Derivation
A combination is a selection of objects from a larger collection of distinct objects, where — unlike a permutation — the order of selection does not matter.
Relationship Between Permutations and Combinations
Sections 3 and 5 derived and independently, but they are not two unrelated formulas — was, in fact, derived directly from by dividing out the internal orderings.
Summary
This chapter developed the basic tools of combinatorics used throughout the rest of the course.
More questions
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- Example 1A restaurant menu offers 4 starters, 6 main courses and 3 desserts. In how many ways can a customer choose one item from each course to make…Free
- Example 2How many 3-digit numbers can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated in a single number?Free
- Example 3If $\dfrac{1}{6!} + \dfrac{1}{7!} = \dfrac{x}{8!}$, find the value of $x$.Free
- Example 4How many 4-letter words (not necessarily meaningful) can be formed using the letters of the word \"ORANGE\" (all 6 letters distinct), withou…Preview
- Example 5How many 3-digit numbers can be formed using the digits $1$ to $9$ if repetition of digits is allowed?Preview
- Example 6In how many ways can 5 boys and 3 girls be seated in a row so that all 3 girls always sit together?Preview
- Example 7In how many different ways can the letters of the word \"ASSASSIN\" be arranged?Preview
- Example 8A committee of 5 is to be formed from a group of 6 men and 4 women. In how many ways can this committee be formed if it must contain at leas…Preview
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- Q9A person owns 5 different pens and 3 different notebooks. In how many ways can he select one pen and one notebook to carry to a meeting?Free
- Q10How many 2-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$ (no digit repeated in a number), given that a number cannot start…Preview
- Q11A fixed-price meal lets a customer choose one of 3 soups or skip the soup course entirely, then choose one of 5 main courses and one of 2 de…Preview
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- Q14Evaluate $^{7}P_{3}$.Free
- Q15How many 5-letter words can be formed using the letters of the word \"NUMBER\" (6 distinct letters), without repeating any letter?Free
- Q16In how many ways can the letters of the word \"MOBILE\" be arranged so that the vowels (O, I, E) always occupy the odd positions (1st, 3rd,…Preview
- Q17How many 4-digit numbers greater than $5000$ can be formed using the digits $1, 3, 5, 7, 9$ if repetition of digits is allowed?Preview
- Q18In how many ways can 5 different books be arranged on a shelf so that two particular books are never together?Preview
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- Q19Evaluate $^{10}C_{4}$.Free
- Q20If $^{15}C_{r} = {}^{15}C_{r+3}$, find the value of $r$.Free
- Q21A cricket team of 11 players is to be selected from a squad of 15 players. In how many ways can this be done?Preview
- Q22From a group of 6 boys and 4 girls, a team of 5 is to be selected so that it contains exactly 3 boys and 2 girls. In how many ways can this…Preview
- Q23A bag contains 5 red balls and 6 black balls. In how many ways can 4 balls be selected so that at least one selected ball is red?Preview