In how many ways can 12 indistinguishable apples and 1 orang

In how many ways can 12 indistinguishable apples and 1 orange be distributed among three children in such a way that each child gets at least one piece of fruit?

Solution

Case 1. Orange is given to some child

Choose 1 child out of three in C(3,1)=3 ways.

Apple is given to other two in 1 way.

Now there are 10 more apples to which can be distributed to each kid given 0 or more apples

The number of ways of distributing n identical objects among r groups such that each group can have 0 or more objects <=n

is: C(n+r-1,r-1)

Hence number of ways of giving n=10 apples to r=3 children where each can get 0 or more apples is

C(10+3-1,3-1)=C(12,2)=12*11/2=66

So total number of ways is: 66*3=198

Case 2. All get only apples

First we give an apple to each children

Now we have 9 apples:n=9 and r=3 children and each can get 0 or more apples. So number of ways to distribute are:L

C(9+3-1,3-1)=C(11,2)=55

So ,55 ways.

So total ways to distribute are:

198+55=253