# Formula n(n-1)/2

I know that n(n+1)/2 is getting the sum of 1 to n numbers.
How about the n(n-1)/2?

where and when do we use this formula? and what other formulas are related to these two?

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Iâ€™d say, that if \frac{n(n+1)}{2} is som of n numbers, then \frac{(n-1)n}{2} is the sum of n-1 numbers, do you agree?

You know, itâ€™s not easy to answer the question without the proper contextâ€¦

Second formula can also be used to find out number of combinations how to choose two elements out of n, or how many elements Ai,j are in square matrix where i < j and probably one can find another dozen of descriptionsâ€¦

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As @betlista has said, n(n-1)/2 is the sum of the first (n-1) numbers, that is

1 + 2 + 3 + 4 + .......... + (n-1)


Now one might think that there is not much use for this formula, but when you do some research, you can find interesting uses for it. @betlista has explained a few uses. Here is a link which explains one usage

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So basically it is a combination formula where you need to choose r element out of n and order does not matter.

n * (n-1) / 2

Same as NCR forumula which is N! / (N-R)! * R!

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Bruh its been six years

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Legend says he is still asking this question. Duhhhh

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Your answer is right , if you look at T(n) for insertion sort here you can see how outer loop runs for n(n+1)/2 -1 times and inner while runs one less than test case which again yields (n-1)(n-1+1)/2=n(n-1)/2

(n + 1) * (n + 2)/ 2 are both are same n*(n + 1) / 2

can anyone explain?

n*(n-1)/2 .This formula is mostly used to find total number of subarrays for a given array of size n

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I think it should have been \frac{n\times (n+1)}{2}. You missed subarrays of size 1.

nC2 ->Combination, number of handshakes

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This link is not working.

It was posted in 2015

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use n as 5
it give sum of n number
n*(n+1)/2 = 5(5+1)/2=15

it give sum of n+1 number
(n+1)(n+2)/2 =(67)/2=31

it give sum of n-1 number
n*(n-1)/2 = 5(5-1)/2=10

it give sum of square n2
n(n+1)(2n+1)/6 = 5(6)(11)/6=55

it give sum of cube n3
[ n*(n-1)/2]2 =( 5(5-1)/2)2=100

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