Question

How many four integer combinations from numbers 1 to 100 equals 100

Answer 1

This is a very complicated questions, unless I mis-understood the question.

I assume you mean something like this:
12 26 8 54 since 12+26+8+54 = 100

would you count 26 8 12 54 as the same combination?
If not, there would be 4! or 24 arrangements of this same combination.

I made up a rather silly computer program and ran it on my MAC

10 for a = 1 to 100
20 for b = 1 to 100
30 for c = 1 to 100
40 for d = 1 to 100
50 sum = a+b+c+d
60 if sum = 100 then count = count+1
70 next d
80 next c
90 next b
100 next a
110 print count

It gave me a total of 156,849 of them!!
(e.g. one of them was 12 26 8 54)
It would include 4! or 24 same permutations for each sum of 100 of the same 4 integers, so we should divide the 156849 by 24

But....
suppose we look at another one of them:
30 30 10 30 , which is part of my 156,849
but there are only 4!/3! or 4 possible arrangements.
Awwhhh, the complications!

btw, my dinky little program had to perform 100 million iterations to find that sum!!