There are n people competing in a chess tournament. Each competitor...

If n people play with every other member exactly once (k = 1) then the total number of matches = nC2 = (n(n- 1))/2

If they have to play with each other k times then the total = k(n^2 - n)/2

Answer: C

These are great solutions. There are still a couple of completely different ways to solve this question.
Any takers?

Cheers,
Brent

Let me assume a student who does not know how to use permutation and combination and yet he wants to solve this questions.

He can use a bit of logic and hit and trial to get the answer.

So suppose if I consider, N = 2 and k = 2, the number of games played should be 2 ( easy to visualize, A and B are two people, they play matches 2 times)

A) kn – k = 4 - 2 = 2
B) (n² – 2k)/2 = 4 -4/2 = 0 = Not possible
C) k(n² – n)/2 = 2.2/2 = 2
D) (n² – 2nk + k)/2 = 4 - 8 +2/2 = negative = Not Possilbe
E) (kn – 2k)/2 = 4- 4/2 = 0 Not possible

Now we are left with, Option A and C.

Take N = 3 and k = 1, if there are three people A, B, C, the matches they can play are AB, BC and AC i.e. three matches.

A. 1.3 - 1 = 2 Not possible
C. 1.(9-3)2/ = 3 Hence C has to be the answer.


Thanks,
Saquib
Quant Expert
e-GMAT

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Here's an approach that doesn't require any formal counting techniques:

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 includes the match played against Competitor B. Likewise, when Competitor B says he/she played n-1 other competitors, this includes 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:

Cheers,
Brent

shouldn't the answer be \(k*\frac{(n^2 – n)}{2}\) and not \(k*\frac{(n^2 – n)}{6}\)
I assume this to be a typo

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