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# counting

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32
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In how many ways can you write 20 as a sum of three positive integers?

1 + 2 + 17 and 17 + 2 + 1 are different ways.

Oct 2, 2020

#1
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Partitions of 20 into 3 parts:

There are 33 such partitions:

18 + 1 + 1 = 20
17 + 2 + 1 = 20
16 + 3 + 1 = 20
16 + 2 + 2 = 20
15 + 4 + 1 = 20
15 + 3 + 2 = 20
14 + 5 + 1 = 20
14 + 4 + 2 = 20
14 + 3 + 3 = 20
13 + 6 + 1 = 20
13 + 5 + 2 = 20
13 + 4 + 3 = 20
12 + 7 + 1 = 20
12 + 6 + 2 = 20
12 + 5 + 3 = 20
12 + 4 + 4 = 20
11 + 8 + 1 = 20
11 + 7 + 2 = 20
11 + 6 + 3 = 20
11 + 5 + 4 = 20
10 + 9 + 1 = 20
10 + 8 + 2 = 20
10 + 7 + 3 = 20
10 + 6 + 4 = 20
10 + 5 + 5 = 20
9 + 9 + 2 = 20
9 + 8 + 3 = 20
9 + 7 + 4 = 20
9 + 6 + 5 = 20
8 + 8 + 4 = 20
8 + 7 + 5 = 20
8 + 6 + 6 = 20
7 + 7 + 6 = 20

Oct 2, 2020
#2
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In "Partitions Theorem",  17 + 2 + 1 are THE SAME as 1 + 2 + 17. If you wish to re-arrange them, then you will have to calculate the "permutations" of each partition. So, 17 + 2 + 1 =3! = 6 permutations. You have to do that for ALL 33 partitions keeping in mind that some of them have duplicates such as 8 + 8 + 4 =6!/2 = 3 permutations.

Oct 2, 2020