Discrete math help (combination of numbers

Question

Question: In how many ways can 15 be written as the sum of 2’s and 3’s if the order of the summands is a) relevant or b) not relevant

Answer 1

How many threes must we have ? 5, 3, 1. (because the remainder has to be even).

So if the order is irrelevant we have

3 3 3 3 3

3 3 3 2 2 2

3 2 2 2 2 2 2

So 3 ways.

If the order is relevant.

3 3 3 3 3 only 1 way here as all digits are 3s

3 3 3 2 2 2 . There are 6 digits so they can be placed in a line in 6! ways. As the 3 is repeated 3 times we have to divide by 3!. As the 2 is repeated 3 times we have to divide by 3!. So 6! / 3!3! = 720 / 36 = 20 ways

3 2 2 2 2 2 2 7 digits, so can be placed in 7! ways. As 2 is repeated 6 times we divide by 6!

so 7!/6! = 7 ways

So if order is important : 1 + 20 + 7 = 28 ways