In this section, the following topics are discussed with examples

**Counting formula for Combinations****Important Rules****Examples**

### COUNTING FORMULA FOR COMBINATION

### 1. Selection of Objects Without Repetition

The number of combinations or selections of n different things taken r at a time is denoted by nCr or C (n, r) where ,

where n is a natural number and r is a whole number.

### SOME IMPORTANT RULES:

## It is the combination of reasonable talent and the ability to keep going in the face of defeat that leads to success.

Sherry Bebitch Jeffe

#### Example 1:

If ^{20}C_{r} = ^{20}C_{r} _{– 10}, then find the value of ^{18}C_{r} .

**Solution:**

^{20}C_{r} = ^{20}C_{r – 10 }⇒ r + (r – 10) = 20 ⇒ r = 1

∴ ^{18}C_{r }= ^{18}C_{15} = ^{18}C_{3}

=

= 816.

#### Example 2

How many different 4-letter words can be formed with the letters of the word ‘JAIPUR’ when A and I are always to be included ?**Solution:**

Since A and I are always to be included, so first we select 2 letters from the remaining 4, which can be done in ^{4}C_{2} = 6 ways. Now these 4 letters can be arranged in 4! = 24 ways, so the required number = 6 × 24 = 144.

### 2. Selection of Objects With Repetition

The total number of selections of r things from n different things when each thing may be repeated any number of times is ^{n + r –} ^{1}C_{r} .

### 3. Restricted Selection

(i) Number of combinations of n different things taken r at a time when k particular things always occur is ^{n – k}C_{r – k} .

(ii) Number of combinations of n different things taken r at a time when k particular things never occur i ^{n – k}C_{r}.

### 4. Selection From Distinct Objects

Number of ways of selecting at least one thing from n different things is^{n}C_{1} + ^{n}C_{2} + ^{n}C_{3} + …..+ ^{n}C_{n} = 2^{n} – 1.

This can also be stated as the total number of combination of n different things is 2^{n} – 1.