You purchased 20 identical toy cars they are not distinguish

You purchased 20 identical toy cars (they are not distinguishable). You have 8 children that you are going to give cars to. You want to be sure that each child gets at least one car, and you want to be sure you have given away all of the cars to these children. How many ways can you distribute the cars?

Solution

Consider 8 children as beggars and first give 1 car to each of the children, so that no children has a zero car

Number of cars left = 20 - 8 = 12

Now using beggar\'s method distributing n identical cars in r childrens is given by

C(n+r-1,r-1) = C(12+8-1,8-1) = C(19,7)

Number of ways = 19!/(12!7!) = 50388 ways