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# counting and probability help

+12
195
9
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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 $$a \leq b \leq c \leq d$$?

Jun 18, 2020

#1
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There's:

1 * 2 * 3 * 6

Jun 18, 2020
#2
+638
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that's not the only way...

I need the full answer and solution

amazingxin777  Jun 18, 2020
edited by amazingxin777  Jun 18, 2020
#3
+995
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do it yourself!

List all the combinations then arrange then to the format!

Guest gave a wonderful hint!

hugomimihu  Jun 18, 2020
#4
+638
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tHe hint guest gave i already knew  but it is a great start!

amazingxin777  Jun 18, 2020
#7
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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
edited by thelizzybeth  Jun 18, 2020
#8
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jimkey17  Jun 18, 2020
#9
+638
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I'll think about complementary counting, haven't tried that method on this problem yet...

Thanks for the idea

amazingxin777  Jun 22, 2020
#5
+311
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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.

Jun 18, 2020
#6
+638
+5

thank you that was correct!!!!

amazingxin777  Jun 18, 2020