Combinations – Formulae with Examples

Combinations

 

Hello Readers

Today’s topic is Combinations (Selection, committee, group)

 

Each of the different groups or selections which can be formed by taking some or all of a number of objects called combinations.

 

Suppose we want to select two out of three persons A, B, C. Then possible selections are AB, BC, CA

Note AB and BA represent the same selection.

 

Suppose we want to select three out of three persons A, B, C. Then possible selections are ABC.

NoteABC and BAC and CAB are the same selection.

 

Number of Combinations –

 

Combinations 1

 

Note –

If (n=r), then nCn = 1 and nCo = 1

 

Important Points:

 

nCr = nC(n-r)

e.g.

16C13 = 16C16-13 = 16C3

 

Combinations 2

Let’s discuss some examples –

 

Example 1

 

Find the value of 5C2

 

Solution –

Combination 3

 

Example 2

 

Find the value of n when nC2 = 105?

 

Solution –

nC2 = 105

Combination 4

 

Example 3

 

There are 15 persons in a group. They hand shake to each other. Find the different number of handshake.

 

Solution –

Combination 5

 

Example – 4

 

From a group of 10 men and 5 women. 4 persons are to be selected to form a committee. Find the different number of ways of selection.

 

Solution –

Combination 6

 

 

Example 5

 

From the group of 10 men and 5 men. 4 persons to be selected such that 3 men and 1 woman to be selected to form a committee. Find the different number of ways of selection.

 

Solution –

 

Combination 7

 

Example 6

 

From a group of 10 men and 5 women. 4 persons are to be selected such that 4 men or 4 women in the group. Find the different number of ways.

 

Solution –

 

Combination 8

 

Example 7

 

A committee of 5 members is to be formed out of 4 men and 5 women.

 

(i) In how many ways can a committee consisting of at least 1 woman be formed?

 

Solution –

 

5C1 × 4C4 + 5C2 × 4C3 + 5C3 × 4C2 + 5C4 × 4C1 + 5C5 × 4C0

= 5 × 1 + 10 × 4 + 10 × 6 + 5 × 4 + 1

= 126

 

(ii) In how many ways can a committee consisting of 3 men and 2 women be formed?

 

Solution –

 

Combination 9

 

With this, we will finish this topic.

 

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