Скачать или смотреть Class 11 Maths Ch 6 Exercise 6.4 solution Permutations and combinations

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By
Ankit Ghevariya
e-mail- [email protected]

*Permutations*
Timestamps:
0:35 Question 1 class 11 maths chapter 6
3:51 Question 2(1)
7:25 Question 2(2)
10:10 Question 3
12:29 Question 4
16:07 Question 5
21:05 Question 6
24:55 Question 7
28:36 Question 8
31:07 Question 9


*Permutations and Combinations*

*1. Introduction*

Permutations and combinations are fundamental concepts in combinatorial mathematics. They deal with counting the number of possible arrangements or selections of items. These concepts are used across various fields including probability, statistics, and computer science.

*2. Permutations*

*Definition:*
A permutation is an arrangement of objects in a specific order. The order of arrangement matters in permutations.

*Formula:*
The number of permutations of \( n \) distinct objects is denoted by \( n! \) (n factorial), which is calculated as:
\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \]

*Example:*
To find the number of ways to arrange the letters in the word "CAT":
The number of permutations is \( 3! = 6 \).
The arrangements are: CAT, CTA, ACT, ATC, TAC, and TCA.

*Permutations of a Subset:*
If you want to find the number of ways to arrange \( r \) objects out of \( n \) distinct objects, the formula is:
\[ P(n, r) = \frac{n!}{(n-r)!} \]

*Example:*
To find how many ways you can arrange 2 out of 3 letters (A, B, C):
\( P(3, 2) = \frac{3!}{(3-2)!} = \frac{6}{1} = 6 \).
The arrangements are AB, AC, BA, BC, CA, and CB.

*3. Combinations*

*Definition:*
A combination is a selection of objects without regard to the order. The order of selection does not matter in combinations.

*Formula:*
The number of combinations of \( n \) distinct objects taken \( r \) at a time is denoted by \( C(n, r) \) or \( \binom{n}{r} \) and is calculated as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

*Example:*
To find how many ways you can choose 2 letters from the set {A, B, C}:
\( \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{6}{2 \times 1} = 3 \).
The combinations are AB, AC, and BC.

*4. Applications*

*Permutations and combinations are applied in various fields:*

*Probability:* To calculate the likelihood of events.
*Statistics:* For determining sample sizes and survey designs.
*Computer Science:* For algorithm design and cryptography.
*Games and Puzzles:* For solving problems related to strategy and game theory.

*5. Key Differences*

*Order Matters:* In permutations, order is important. In combinations, order is not.
*Permutations vs. Combinations Formula:*
Permutations: \( P(n, r) = \frac{n!}{(n-r)!} \)
Combinations: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

*6. Special Cases*

*Permutations with Repetition:* When elements can be repeated, the formula is \( n^r \), where \( n \) is the number of choices and \( r \) is the number of items.
*Combinations with Repetition:* Also known as multisets, the formula is \( \binom{n+r-1}{r} \), where \( n \) is the number of types and \( r \) is the number of items to choose.

*7. Examples in Practice*

*Classroom Seating:* Arranging students in different ways.
*Lottery Numbers:* Choosing a set of numbers where order doesn’t matter.
*Password Creation:* Generating passwords with specific rules for order and repetition.

*8. Conclusion*

Understanding permutations and combinations is crucial for solving many real-world problems involving arrangement and selection. Mastery of these concepts allows for more accurate calculations in probability, effective planning in various fields, and better problem-solving skills overall.

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