At the end of the day a bakery gives everything that is unso

At the end of the day, a bakery gives everything that is unsold to food banks for the needy. If it has 12 apple pies left at the end of a given day, in how many different ways can it distribute these pies among six food banks fat the needy? With reference to Exercise 1.55, in how many different ways can the bakery distribute the 12 apple pies if each of the six food banks is to receive at least one pie?

Solution

1.55

Using the formula, (N+R-1)C(R-1) for ways to put N objects put into R containers:

(12+6-1)C(6-1) = 17C5 = 17!/(5!*12!) = 6188

answer: 6188 ways

1.56

Using the formula, (N-1)C(R-1), for ways to put N objects put into R containers. where all containers must get at least one object:

(12-1)C(6-1) = 11C5 = 11!/(5!*6!) = 462

answer: 462 ways