While going through an equation today i realized that sum of first (n-1) numbers is [n*(n-1)/2] which is equal to combinations of two items out of n i.e [n!/((n-2)! * 2!)]. I need some intuition on how these two things are related?
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You're asking why the number of ways to pick 2 cards out of a deck of
The reason is that there are (n-1) ways to pair the first card with another card, plus (n-2) ways to pair the second card with one of the remaining cards, plus (n-3) ways to pair the third card...
One way to think of this is to map it to an intermediary representation - namely, a triangle made of boxes:
Let's suppose that this triangle has width and height n. Its area is equal to 1 + 2 + 3 + ... + n = n(n+1) / 2.
We can interpret this triangle as every way of choosing two elements out of (n + 1) by expanding it out into a 0/1 matrix:
If we take all unordered pairs of two numbers, then we can always sort the pair by putting the bigger number first. Each possible way to do this corresponds to a 1 entry in this matrix. For example, the pairing (n+1, n) corresponds to the upper-right corner, since n+1 > n. Similarly, (n+1, 1) corresponds to the top-left corner. If you count up the number of 1s in this matrix, you'll note that it's half the area of the matrix. There are n + 1 rows and n columns, so the area is n(n + 1) / 2. We can also arrive here by noting that there are n 1s in the first row, then n - 1, then n - 2, ..., then 1. Thus 1 + 2 + ... + n is equal to the number of unordered pairs drawn from (n + 1) numbers.
Hope this helps!