Statistics · Class 11 Commerce
Ch 6Permutations, Combinations and Binomial Expansion — Class 11 Statistics, concept-first.
Before we can count how many ways an event can happen, we need a systematic way of counting instead of listing every possibility by hand. This is exactly the gap the Fundamental Principle of Counting fills, and it is the foundation on which both permutations and combinations are built — the Gujarat Std-11 Statistics sy…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Combinations
A combination is an unordered selection of objects — and are the same combination, unlike a permutation. The number of ways of choosing objects (order irrelevant) from distinct objects is denoted (" choose ").
Most relevant Q&A
- A committee of 4 members is to be formed from 6 boys and 4 girls, such that the committee has exactly 2 boys and 2 girls. In how many ways c…Free
- How many diagonals can be drawn in a hexagon?Free
- Verify that $^{10}C_{4} = {}^{10}C_{6}$.Preview
- A Std 11 Commerce student in Gujarat has to choose 3 optional subjects out of 5 offered by the school. In how many ways can the student make…Preview
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
Before we can count how many ways an event can happen, we need a systematic way of counting instead of listing every possibility by hand.
Factorial Notation
Repeatedly applying the multiplication principle to arrange a set of distinct objects produces the same kind of product again and again — .
Permutations
Once objects can be told apart and their order matters, arranging them is called a permutation. This is the key word to watch for in a word problem: if changing the order of the same objects counts as…
Combinations
In many real situations, we care only about which objects are chosen, not the order in which they are picked — forming a committee, selecting subjects, or picking a hand of cards.
Binomial Theorem for Positive Integral Index
Multiplying out directly for a large by repeated multiplication is slow and error-prone. The Binomial Theorem gives a direct formula for every term of the expansion, and the coefficients that appear a…
Exercises
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- Q7A committee of 4 members is to be formed from 6 boys and 4 girls, such that the committee has exactly 2 boys and 2 girls. In how many ways c…Free
- Q8How many diagonals can be drawn in a hexagon?Free
- Q9Verify that $^{10}C_{4} = {}^{10}C_{6}$.Preview
- Q12Find the middle term(s) in the expansion of $\left(x + \dfrac{1}{x}\right)^{8}$.Preview
- Q13A Std 11 Commerce student in Gujarat has to choose 3 optional subjects out of 5 offered by the school. In how many ways can the student make…Preview
More questions
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- Example 1A student going to college has 3 different shirts and 4 different trousers. In how many ways can the student dress up by choosing one shirt…Free
- Example 2How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated?Free
- Example 3Evaluate $\dfrac{8!}{6!\,2!}$.Free
- Example 4In how many ways can the letters of the word 'MONDAY' be arranged, taken all at a time?Preview
- Example 5In how many ways can the letters of the word 'COMMITTEE' be arranged, taken all at a time?Preview
- Example 6In how many ways can 1st, 2nd and 3rd prizes be distributed among 8 competitors, if no competitor can receive more than one prize?Preview
- Example 10Expand $(x+2)^4$ using the Binomial Theorem.Preview
- Example 11Find the 4th term in the expansion of $(2x - y)^7$.Preview