Permutations/Combinations

sparkles3144 wrote:There are 16 teams in a tournament. If during the first round, each team plays every other team exactly once, how many games will be played in the first round?

a)15
b)30
c)120
d)240
e)256

Here are two approaches:

Approach #1: Combinations
The question is really asking, "In how many different ways can we create 2-team pairings from 16 teams?"
Since the order of the selections does not matter (i.e., selecting teams A and B to play, is the same as selecting teams B and A to play), we can use combinations.
There are 16 teams and we want to select 2.
This can be accomplished in 16C2 ways (120 ways)

If anyone is interested, we have a free video on calculating combinations (like 16C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789


Approach #2: Ask each team
Let's have every team play every other team exactly once. Then we'll go to a team (say Team A) and ask, "How many different teams did you play?"
Team A's answer will be 15

Then go to another team (say Team B) and ask, "How many different teams did you play?"
Team B's answer will be 15

and so on . . .

Every team (of the 16 teams) will answer 15.
So, (16)(15) = 240
IMPORTANT: There's some duplication here.
For example, when Team A said that it played 15 other teams, it was including the game it played against Team B. When Team B said that it played 15 other teams, it was including the game it played against Team A. So, in our calculation of 240 games, we included the A vs B game twice.
In fact, we counted every game two times.

So, to account for this duplication, we'll take 240 and divide by 2 to get 120

Answer: C

Cheers,
Brent

sparkles3144 wrote:There are 16 teams in a tournament. If during the first round, each team plays every other team exactly once, how many games will be played in the first round?

a)15
b)30
c)120
d)240
e)256

Since there are 16 teams and each team plays every other team once, the number of games played in the first round is 16C2 =16!/[2!(16-2)!] = (16 x 15)/2! = 8 x 15 = 120 games.

Answer: C