In some problems of occupancy theory we are concerned with t

In some problems of occupancy theory we are concerned with the number of ways in which certain indistinguishable objects can be distributed among individuals, urns, boxes, or cells with at least one in each cell. Find an expression for the number of ways in which r indistinguishable objects can be distributed among n cells with at least one in each cell, and rework the numerical part of Exercise 1.9 with each of the three customers getting at lead one loaf of bread. A shipment of 10 television sets includes three that are defective. In how many ways can a hotel purchase four of these sets and receive at lead two of the defective sets?

Solution

0 x 0 x 0 x 0 x 0

there are (5-1) positions for dividers, and we need to place only (3-1) dividers to divide the loaves to 3 customers
0 x 000 x 0 ,eg indicates 1 loaf to 1st customer, 3 to 2nd & 1 to 3rd

now the dividers can be placed in any of C(5-1, 3-1) ways,
and generalising for r loaves & n customers, the formula is
# of ways = C(r-1,n-1)