First slide
Combinations
Question

In how many ways can 20 oranges be given to four children if each child should get at least one orange?

Moderate
Solution

The maximum number of oranges that a child can get = 20 – 3 = 17.

Thus, the problem is equivalent to finding the number of integral solutions to the equation

x1+x2+x3+x4=20

where 1x1,x2,x3,x417, and x1, x2, x3, x4 denote the number of oranges given to the four children.

Hence, the required number of ways is

= coefficient of x20 in x+x2+x3++x174= coefficient of x16 in 1+x+x2++x164= coefficient of x16 in 1x174(1x)4= coefficient of x16 in (1x)4=16+41C16=19C3=969

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