Variables in the answer choices question - chess tournament

[Brent@GMATPrepNow]

There are n people competing in a chess tournament. Each competitor must play every other competitor k times. If n > 1 and k > 0, what is the total number of games played in the tournament?

A) kn - k
B) (n² - 2k)/2
C) k(n² - n)/2
D) (n² - 2nk + k)/2
E) (kn - 2k)/2

Answer: C

Source: www.gmatprepnow.com
Difficulty level: 650

[Brent@GMATPrepNow]

There are n people competing in a chess tournament. Each competitor must play every other competitor k times. If n > 1 and k > 0, what is the total number of games played in the tournament?

A) kn - k
B) (n² - 2k)/2
C) k(n² - n)/2
D) (n² - 2nk + k)/2
E) (kn - 2k)/2

Answer: C

Source: www.gmatprepnow.com
Difficulty level: 650

Here's one approach (there are several others):

Let's say a MATCH is when two competitors sit down to play their k games against each other.

If we ask each of the n competitors, "How many MATCHES did you have?", the answer will be n-1, since each competitor plays every other competitor, but does not play against him/herself.

So, n(n-1) = the total number of MATCHES

IMPORTANT: There's some duplication here.
For example, when Competitor A says that he/she played n-1 other competitors, this included the match played against Competitor B. Likewise, when Competitor B said he/she played n-1 other competitors, this included the match played against Competitor A.

So, in our calculation of n(n-1) = the total number of MATCHES, we included the A vs B match twice.
In fact, we counted every match two times.

To account for this duplication, we'll take n(n-1) and divide by 2 to get n(n-1)/2 MATCHES.

Since each match consists of k games, the total number of games = kn(n-1)/2

Check the answer choices....not there!
However, we can take kn(n-1)/2 and rewrite it as k(n² - n)/2

Answer: C

Cheers,
Brent