How many arrangements of the prime factors of 7560 are possible if they are written out in full. Of these, how many have the factors arranged in ascending order

Question

I have been stuck on this for a while now 

Answer 1

Since the order of factors matter, this is a permutation problem.

 

First, we need to factor out 7560.

7560 = 2³ • 3³ • 5 • 7.

 

There is only one ascending order, and it is shown above.

 

We are ready to find the total number of different permutations of the factors.

 

We use the formula for number of permutations with repeating items. The factor 2 is repeated 3 times, the factor 3 is repeated 3 times, and the other factors are unique.

 

So, the number of permutations is

 

Permutations = 8!/(3! • 3!)

                    =1120.

 

The total number of ways we can arrange the 8 factors of 7560 is 1120. Only one of these arrangements is ascending. In each other arrangements, at least one larger factor will be multiplied by before multiplying one of the smaller factors.