Permutation and combination & Binomial Theorem Revision Notes

 

Permutations, Combinations, and Binomial Theorem: Detailed Notes

Permutations, combinations, and the binomial theorem are key concepts in combinatorics and probability theory. They help solve problems related to counting, arrangements, selections, and expansions, which are fundamental in areas such as probability, statistics, algebra, and discrete mathematics.

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5.1 Value of $nP_r$ and $nC_r$ and Formula-based Problems

Permutations ( $nP_r$ ):

A permutation is an arrangement of objects in a specific order. The number of ways to arrange $r$ objects out of $n$ distinct objects is called a permutation.

Formula for Permutation:

$$nP_r = \frac{n!}{(n - r)!}$$

Where:

• $n$ is the total number of objects.
• $r$ is the number of objects to be selected and arranged.
• $n!$ (read as "n factorial") is the product of all integers from $1$ to $n$: $n! = n(n - 1)(n - 2)\cdots 1$.

Example: Find the number of ways to arrange 3 people out of 5:

$$nP_r = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2!}{2!} = 5 \times 4 \times 3 = 60$$

So, there are 60 ways to arrange 3 people out of 5.

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Combinations ( $nC_r$ ):

A combination is a selection of objects where the order does not matter. The number of ways to choose $r$ objects from $n$ distinct objects is called a combination.

Formula for Combination:

$$nC_r = \frac{n!}{r!(n - r)!}$$

Where:

• $n$ is the total number of objects.
• $r$ is the number of objects to be selected.
• $r!$ and $(n - r)!$ are factorials of $r$ and $(n - r)$, respectively.

Example: Find the number of ways to select 3 people out of 5:

$$nC_r = \frac{5!}{3!2!} = \frac{5 \times 4 \times 3!}{3! \times 2!} = \frac{5 \times 4}{2 \times 1} = 10$$

So, there are 10 ways to select 3 people from 5.

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Formula-based Problems:

1. Permutation of Identical Objects: If some objects are identical, the number of permutations is given by:

   $$nP_r = \frac{n!}{p_1! p_2! \cdots p_k!}$$

   Where $p_1, p_2, \dots, p_k$ are the frequencies of the identical objects.

2. Combination with Restrictions: If there are restrictions on selection, such as "at least one object must be selected," modify the formula accordingly.

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5.2 Problems Based on General Term of a Binomial Expansion

Binomial Theorem:

The binomial theorem provides a formula for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The expansion involves terms of the form $\binom{n}{r} a^{n-r} b^r$, where $\binom{n}{r}$ is the binomial coefficient.

Binomial Theorem Formula:

$$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$

Where:

• $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ is the binomial coefficient.
• $r$ is the term number, starting from 0.
• $a^{n-r}$ and $b^r$ are the powers of $a$ and $b$, respectively.

General Term in Binomial Expansion:

The general term (or $r$-th term) in the expansion of $(a + b)^n$ is given by:

$$T_r = \binom{n}{r} a^{n-r} b^r$$

Where:

• $r$ is the term number.
• $\binom{n}{r}$ is the binomial coefficient.

Example 1: Find the General Term of the Expansion of $(x + 2)^5$

We want to find the general term in the expansion of $(x + 2)^5$.

1. The binomial expansion of $(x + 2)^5$ is:

   $$(x + 2)^5 = \sum_{r=0}^{5} \binom{5}{r} x^{5-r} 2^r$$

2. The general term is:

   $$T_r = \binom{5}{r} x^{5-r} 2^r$$

   Where $r$ varies from 0 to 5.

• For $r = 0$, $T_0 = \binom{5}{0} x^5 2^0 = x^5$.
• For $r = 1$, $T_1 = \binom{5}{1} x^4 2^1 = 5 \cdot x^4 \cdot 2 = 10x^4$.
• For $r = 2$, $T_2 = \binom{5}{2} x^3 2^2 = 10 \cdot x^3 \cdot 4 = 40x^3$.
• For $r = 3$, $T_3 = \binom{5}{3} x^2 2^3 = 10 \cdot x^2 \cdot 8 = 80x^2$.
• For $r = 4$, $T_4 = \binom{5}{4} x^1 2^4 = 5 \cdot x \cdot 16 = 80x$.
• For $r = 5$, $T_5 = \binom{5}{5} x^0 2^5 = 1 \cdot 32 = 32$.

Thus, the expansion is:

$$(x + 2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32$$

Example 2: Find the General Term in the Expansion of $(3x - 2y)^6$

To find the general term in the expansion of $(3x - 2y)^6$, use the binomial theorem:

$$T_r = \binom{6}{r} (3x)^{6-r} (-2y)^r$$

Simplifying the general term:

$$T_r = \binom{6}{r} 3^{6-r} x^{6-r} (-2)^r y^r$$

Thus, the general term is:

$$T_r = \binom{6}{r} 3^{6-r} (-2)^r x^{6-r} y^r$$

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Summary of Key Concepts and Formulas:

1. Permutations:

   $$nP_r = \frac{n!}{(n - r)!}$$

2. Combinations:

   $$nC_r = \frac{n!}{r!(n - r)!}$$

3. Binomial Theorem:

   $$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$

4. General Term in Binomial Expansion:

   $$T_r = \binom{n}{r} a^{n-r} b^r$$

5. Important:

   • Permutations are used when the order matters.
   • Combinations are used when the order does not matter.
   • The general term formula in the binomial expansion allows for the calculation of any specific term in the expansion of $(a + b)^n$.

These concepts are crucial for solving a wide range of problems in combinatorics, probability, and algebra.