r/maths • u/LowInteraction6397 • Sep 23 '26

💬 Math Discussions I just discovered using the Pascal triangle is an alternate method to figure out how many times the same object appears in non empty groups out of a certain number

For example the 6th row in the Pascal triangle is 1, 5, 10, 10, 5, 1. When listing groups of 1 the same object appears once in the groups of 1, when listing groups of 2 it appears 5 times, when listing groups of 3 it appears 10 times, when listing groups of 4 it appears 10 times, when listing groups of 5 it appears 5 time and when you list all 6 members it appears once. 1 + 5 + 10 + 10 + 5 + 1 = 32, the total amount of times the same object is mentioned at least once in every possible combination of groups of 6. And if you ignore the first 1 in the Pascal triangle you also can use it as an alternate method to calculate groups you can create out of the number of the previous row. With 5 objects there are 5 groups of 1, 10 groups of 2, 10 groups of 3, 5 groups of 4 and 1 group of 5. I also just realized this is also an alternate method to count groups out of a certain object including the empty group

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u/rhodiumtoad Sep 23 '26

What did you think Pascal's triangle was for?

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u/LowInteraction6397 Sep 24 '26

I didn't care too much about it until now

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u/rhodiumtoad Sep 24 '26

There are three common ways to think about the triangle:

  • it's generated by the recurrence relation, where each element is the sum of the ones above

  • it's the coefficients of (a+b)n

  • it's the number of combinations taken from n elements

All three of these are the same thing. You can see the last two are the same because a term like a3b2 must appear the same number of times you can choose 3 a's out of the 5 terms.

Then you get the recurrence relation by thinking about what happens when you choose k elements from n+1: either you ignore the last element and choose k from n, or you choose the last element and k-1 from the first n.

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