Question

A. How many ways can you distribute $4$ different balls among $4$ different boxes? B. How many ways can you distribute $4$ identical balls among $4$ identical boxes? C. How many ways can you distribute $4$ identical balls among $4$ different boxes? NOTE QUESTIONS ARE ALL DIFFERENT

Answer 1

Answer:

The answer is explained below

Step-by-step explanation:

A. How many ways can you distribute 4 different balls among 4 different boxes

The number of ways in which 4 different balls can be distributed among 4 different boxes is $4^4=256\ ways$

B. How many ways can you distribute 4 identical balls among 4 identical boxes?

The number of ways in which 4 identical balls can be distributed among 4 identical boxes = P(4,1) + P(4,2) +P(4,3) + P(4,4) = 1 + 2 + 1 + 1 = 5 ways

Where P(k,n) is the number of partitions that k can be divided into n parts

P(4,1) = 4 = 1

P(4,2) = 1 + 3, 2+2 = 2

P(4,3) = 1 + 1 + 2 = 1

P(4,4) = 1 + 1 + 1 + 1 = 1

C. How many ways can you distribute 4 identical balls among 4 different boxes

The number of ways in which 4 identical balls can be distributed among 4 different boxes = $C(4\ ball+4\ box-1,4\ box-1)=C(7,3)=\frac{7!}{(7-3)!3!}=35\ ways$