Meaning of combination is selection of objects.

Selection of objects without repetition:

The number of selections (combinations or groups) that can be formed form n different objects taken r(0 < r < n) at a time is nCr = n!/(r!(n-r))!.

Explanation: Let the total number of selection (or groups) = x. Each group contains r objects, which can be arranged in r! ways. Hence number of arrangements of r objects = x × (r!).

But the number of arrangements = npr

=> x × (r!) = npr =>  x =  => x = n!/(r!(n-r)!) = nCr.

Selection of objects with repetition

The number of combination of n distinct objects taken r at a time when each may occur once, twice, thrice, ...... upto r times, in any combination = n+r-1Cr.

Explanation: Let eh n objects a1, a2, a3, ...... an. In a particular group of r objects, let

        a1 occur x1 times,

        a2 occur x2 times,

        a3 occur x3 times,

        .................................

        .................................

        an occur xn times,

such that x1 + x2 + x3 + ......... + xn = r                   ......... (1)

0 < xi < r   i ε {1, 2, 3, ......, n}.

Now the total number of selections of r objects, out of n

= number of non-negative integral solution of equation (1)

= n+r-1Cn-1 = n+r-1Cr.

 Note:     Details of finding the number of integral solutions of equation (1) are given on page 12 (Multinomial theorem).

Illustration:

Let 15 toys be distributed among 3 children subject to the condition that any child can take any number of toys. Find the required number of ways to do this if

(i)     toys are distinct,                 (ii)    if toys are identical

Solution:

(i)     Toys are distinct

Here we have 3 children and we want the 15 toys to go to the 3 children with repletion. In other words it is same as selecting and arranging children 15 times out of 3 children with the condition that any children can be selected any no. of time which can be done in 315 ways (n = 3, r = 15).

(ii)    Toys are identical

Here we only have to select children 15 times out of 3 children with the condition that any children can be selected any number of times which can be done in

3+5-1C15 = 17C2 way (n = 3, r = 5).