This GSEB Class 11 Commerce Statistics Notes Chapter 6 Permutations, Combinations and Binomial Expansion covers all the important topics and concepts as mentioned in the chapter.

## Permutations, Combinations and Binomial Expansion Class 11 GSEB Notes

**Basic Principle of Counting:**

- Basic Principle of Counting for Addition: If there are m distinct things in Group 1 and n distinct things in Group 2, then selection of one thing from total things of both groups can be done in m + n ways.
- Basic Principle of Counting for Multiplication: If the first operation can be done in m ways and corresponding to it second operation can be done in n ways, then two operations together can be done in m × n ways.
- General form of basic principle: If the first operation can be done in m ways, following which the second operation can be done in n ways and following which the third operation can be done in p ways then the total number of ways of occurrence of three operation together is m × n × p.

**Meaning of Permutation:**

If r distinct things out of given n distinct things are to arranged in r (1 ≤ r ≤ n) different places, then each such arrangement is called a permutation. The total number of such arrangements is denoted by ^{n}p_{r} .^{n}p_{r} p(n, r)

- Order of the things is very important in permutation. AB and BA are two different permutations.
^{n}P_{r}= n(n – 1) (n – 2) … (n – r + 1); 1 ≤ r ≤ n

**Factorial:**

The multiplication of numbers 1 to n, in their respective order, is called n factorial. It is denoted by n!.

- n! = n × (n – 1) × (n – 2)×… ×3 × 2 × 1
- 0! = 1 and 1! = 1

In factorial form: nPr = \(\frac{n !}{(n-r) !}\); n > 0, r ≥ 0, n ≥ r. n and r positive integers.

Some Results:

^{n}P_{0} = 1, ^{n}P_{n} = n !, ^{n}P_{1} = n, ^{n}P_{n-1} = n!

**Permutations of identical things:**

Prom n things, if p is the first type of identical things, q is the second type of identical things and r is the third type of indentical things then arrangement of n things are called permutations of identical objects.

Here, total number of permutation for n things = \(\frac{n !}{p ! q ! r !}\)

**Meaning of Combination:**

The total number of selection of r (r ≤ n) things out of n different things is called combination. It is denoted by ^{n}C_{r}

- Here order is ignored. AB and BA are considered as same selection.
^{n}C_{r}= \(\frac{n !}{r !(n-r) !}\)^{n}C_{0}= 1;^{n}C_{n}= 1;^{n}C_{r}=^{n}C_{n-r}

^{n}C_{1}= n,^{n}C_{n-1}= n- If
^{n}C_{x}=^{n}C_{y}, then x + y = n OR x = y.

**Meaning of Binomial Expansion:**

- Binomial Expression: Any expression consisting of two terms connected by + or – sign is called binomial expression.
- Binomial Expansion: The expansion of binomial expression (x + a) with positive integer power n is called binomial expansion, which is as follows:
- (x + a)
^{n}=^{n}C_{0}x^{n}+^{n}C_{1}x^{n-1}a +^{n}C_{2}x^{n-2}a^{2}+^{n}C_{3}x^{n-3}a^{3}+ … +^{n}C_{n}a^{n} - General terms of binomial expansion is T
_{r+1}=^{n}Cr x^{n-r}. a^{r}

Characteristic of Binomial Expansion:

- The total number of terms in binomial expansion is (n + 1).
- In successive terms of expansion, the power of x keeps reducing by one while power of a keeps increasing by one.
- In any term of expansion, the sum of power of x and a is n.
- The coefficients of successive terms of expansion are
^{n}C_{0},^{n}C_{1},^{n}C_{2}……..^{n}C_{n}. - The coefficients of terms, placed equidistant from the midterm, are equal.

**Important Results**

1. Permutations:

^{n}P_{r}= n(n- 1) (n-2)… (n-r + 1)^{n}P_{r}= \(\frac{n !}{(n-r) !}\)^{n}P_{n}= n !- n ! = n(n- 1) (n-2) (n-3) … × 3 × 2 × 1
- Identical permutations :

Total permutations of ‘n’ things = \(\frac{n !}{p ! q ! r !}\)

2. Combination:

^{n}Cr = \(\frac{n !}{r !(n-r) !}\)^{n}C_{0}= 1,^{n}C_{1}= n,^{n}C_{n-1}= n^{n}C_{n}= 1^{n}C_{r}=^{n}C_{n-r}- If
^{n}Cx =^{n}Cy. then x + y = n or x = y ^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}^{n}C_{0}+^{n}C_{1}+ … +^{n}C_{n }= 2^{n}

3. Binomial expansion:

- (x + a)
^{n}=^{n}C_{0}x^{n}+^{n}C_{1}x^{n-1}a +^{n}C_{2}x^{n-2}a^{2}+^{n}C3x^{n-3}a^{3}+ … +^{n}Cna^{n} - (x – a)
^{n}=^{n}C_{0}x^{n}–^{n}C_{1}x^{n-1}a +^{n}C_{2}x^{n-2}a^{2}–^{n}C3x^{n-3}a^{3}+ … ±^{n}Cna^{n}

4. General term of Binomial expansion:

General term = ^{n}C_{r} x^{n-r} ar.

This term is (r + 1)th term of binomial expansion.

5. Some results:

- If n is even:

The middle term of the expansion (x + a)^{n}is obtained by putting r = \(\frac{n}{2}\) in the general term. - If n is odd:

Two middle terms of the expansion of (x + a)^{n}is^{n}C_{0}+^{n}C_{1}+^{n}C_{2}+ ……… +^{n}C_{n}= 2^{n}

6. Paseals triangle for coefficients of Binomial expansion: