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

Question

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

*kudos for all correct solutions

EgmatQuantExpert · Accepted Answer

• If each player plays with each other only once then the total number games is simply given by
 o  ^nC_2 = \frac{n!}{ } = \frac{ }{2} 
o Where n is the number of players and we choose 2 players at a time to play a single game. 
• But then each player is playing its competitor “k” times. 
 o Hence the total number of games would be multiplied “k” times.
o Thus the total number of games = k * Total number of games played once  = k*n*\frac{(n-1)}{2} = k*\frac{(n^2 – n)}{6}  
• Hence the correct answer is Option C.

BrentGMATPrepNow · Answer

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 =  k   n   (n-1)/2

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

Answer:  C

Cheers,
Brent

Kurtosis · Answer

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

BrentGMATPrepNow · Answer

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

Cheers,
Brent

Mo2men · Answer

I will do this logically

If I choose 3 teams to meet only 1 round. then total number of games =3

n= 3 & k=1 & total =3

Plug in the answer choices

A) kn – k ..................................3-1=2.............................Eliminate

B) (n² – 2k)/2......................... (9-2)/2.............................Eliminate

C) k(n² – n)/2...........................1 * (9-3)/2=3....................Keep

D) (n² – 2nk + k)/2..................(9-6+1)/2.........................Eliminate

E) (kn – 2k)/2..........................(3-2)/2.............................Eliminate

Answer: C

EgmatQuantExpert · Answer

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 Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

https://e-gmat.com/blogs/wp-content/uploads/2016/12/NP-Free-Trial-e1481107541356.png

storm1327 · Answer

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

luisdicampo · Answer

/2 not 6, thanks by the way, very helpful!

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