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?
Intuition on the sum of first (n-1) numbers is equal to the number of ways of picking 2 items out of n.
4 Answers
Let's count 2-subsets from a set of size $n$.
Certainly this is $n\choose2$.
Alternatively, we can count by cases. First some details:
Let $N$ be the set $\{x_1, x_2, \ldots x_{n}\}$ where $x_i < x_{i+1}$ for $1 \leq i \leq n - 1$.
We seek to count how many unique subsets $\{a, b\} \subseteq N$ there exist where $a < b$.
Cases:
1) $b = x_n$ $\hspace{14mm}$ As $a < b$, there remain $(n - 1)$ choices for a.
2) $b = x_{n-1}$ $\hspace{9mm}$ There remain $(n - 2)$ choices for $a$.
3) $b = x_{n-2}$ $\hspace{9mm}$ There remain $(n - 3)$ choices for $a$.
$\hspace{10mm}\vdots$
$n$) $b = x_0$ $\hspace{13mm}$ There remain $0$ choices for $a$.
Summing over each of these distinct cases gives us $\sum_{i=0}^{n-1}i$.
Hence, $\frac{n(n-1)}{2} =$ $n\choose2$ $= 1 + 2 + \ldots (n-1)$.
You're asking why the number of ways to pick 2 cards out of a deck of n
is the same as the sum 1 + 2 + ... + (n-1).
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...
Wolfram proof without words. Note that this uses Pascal's Triangle.
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0Sorry guys, the link to wolfram gives a good intuition but the explanation above is a bit misleading, so i have removed it and selected this as the accepted answer. – 2012-08-23
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:
| 1 2 ... n-1 n ----+------------------ n+1 | 1 1 1 1 1 n | 1 1 1 1 0 n-1 | 1 1 1 0 0 ... | 1 1 0 0 0 2 | 1 0 0 0 0 1 | 0 0 0 0 0
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!