In how many ways can 36 be written as the product \(a \times b \times c \times d\) , where \(a, b, c\) and \(d\) are positive integers such that \(a \leq b \leq c \leq d\)?

amazingxin777 Jun 18, 2020

#1

#3**-2 **

do it yourself!

List all the combinations then arrange then to the format!

Guest gave a wonderful hint!

hugomimihu
Jun 18, 2020

#7**+1 **

After solving the problem, I'm not really sure how this hint ties into the problem.

Are you saying there is 1 option for a, 2 options for b, 3 options for c, and 6 options for b? Or did you factorize 36?

Just curious to see if your solution was maybe faster than casework.

thelizzybeth
Jun 18, 2020

#9**+4 **

I'll think about complementary counting, haven't tried that method on this problem yet...

Thanks for the idea

amazingxin777
Jun 22, 2020

#5**+1 **

There aren't that many cases because of the \(a \le b \le c \le d\) restriction, so we don't have to worry about permutations.

Using casework we find:

**Case 1: 3 ones (36 as 1 factor)**

1*1*1*36

**Case 2: 2 ones (36 as 2 factors)**

1*1*2*18

1*1*3*12

1*1*4*9

1*1*6*6

**Case 3: 1 one (36 as 3 factors)**

1*2*3*6

1*2*2*9

1*3*3*4

**Case 4: 0 ones (36 as 4 factors)**

2*2*3*3

Counting them up, we find that there are 9.

thelizzybeth Jun 18, 2020