Kenneth Rosen Edition 7 Exercise 6.5 Question 24 (Page No. 432)

Question

How many ways are there to distribute $15$ distinguishable objects into five distinguishable boxes so that the boxes have one, two, three, four, and five objects in them, respectively.

Answer 1

Let boxes be A,B,C,D,E . And 15 distinguishable objects are there .

So, the given problem is a type of DODB problem.

Suppose for a case when boxes have following configuration  A- 1, B- 2 , C- 3, D -4 , E-5

Then number of combinations for this particular case is 15C1 * 14C2 * 12C3 * 9C4 * 5C5 = 37837800

Now we know we have total 5! different configurations of boxes. So total ways or combinations possible = 5! * 37837800=4540536000

Answer 2

A,B,C,D,E (boxes)

1 2 3 4 5

firstly assume the objects are indistinguishable,

so they can be distributed in 5! = 120 ways = X

but since they are actually distinguishable, they can be permutated for any pattern of distribution in

15!/5!4!3!2! = 37837800 ways = Y

 

so total ways = XY = 4540536000