Combination:- Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

Examples:

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

Note: AB and BA represent the same selection.

All the combinations formed by a, b, c taking ab, bc, ca.

The only combination that can be formed of three letters a, b, c taken all at a time is abc.

Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

Note that ab ba are two different permutations but they represent the same combination.

Number of Combinations:

The number of all combinations of n things, taken r at a time is:

nCr = n! = n(n - 1)(n - 2) ... to r factors .

(r!)(n - r)! r!

ⁿC_r = n!/r!(n-r)!

Note:

ⁿC_n = 1 and ⁿC₀ = 1.

ⁿC_r = ⁿC_{(n - r)}

Examples:

i.   ¹¹C₄ = (11 x 10 x 9 x 8)/(4 x 3 x 2 x 1) = 330

ii.   ¹⁶C₁₃ = ¹⁶C_{(16 - 13)} = ¹⁶C₃ = 16 x 15 x 14/3! = 16 x 15 x 14/6 = 560.

Number of handshakes:-

Suppose there are N people. Each person shakes hands with every other person. Since there are N-1 other people, each person shakes hands with N-1 other people. So there are N(N-1) total cases of a hand shaking. Since each handshake has two hands shaking, divide the number of hands shaking by two to get the number of handshakes.

or

Number of handshakes of n person is  =  ⁿC₂