Problem Solving

AAPL wrote:Veritas Prep

There are 16 teams in a soccer league, and each team plays each of the others once. Given that each game is played by two teams, how many total games will be played?

A. 256
B. 230
C. 196
D. 169
E. 120

OA E

There are 16 teams. If we ask each team, "How many teams did you play?" we'll find that each team played 15 teams, which gives us a total of 240 games (since 16 x 15 = 240).

From here we need to recognize that each game has been COUNTED TWICE.
For example, if Team A and Team B play a game, then Team A counts it as a game, and Team B ALSO counts it as a game.

So, to account for the DUPLICATION, we'll divide 240 by 2 to get 120

Answer: E
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Here are two related questions:
https://www.beatthegmat.com/ugghhh-i-pi ... 67675.html
https://www.beatthegmat.com/number-of-h ... 00109.html

Cheers,
Brent

AAPL wrote:Veritas Prep

There are 16 teams in a soccer league, and each team plays each of the others once. Given that each game is played by two teams, how many total games will be played?

A. 256
B. 230
C. 196
D. 169
E. 120

OA E

Another approach:
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)

Answer: E

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

Cheers,
Brent

AAPL wrote:Veritas Prep

There are 16 teams in a soccer league, and each team plays each of the others once. Given that each game is played by two teams, how many total games will be played?

A. 256
B. 230
C. 196
D. 169
E. 120

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

Alternate Solution:

Let's see the pattern that develops:

The first team plays each of the 15 teams besides itself..

The second team has already been paired with the first team, so it plays each of the remaining 14 teams.

The third team has already been paired with the first two teams, so it plays each of the remaining 13 teams.

Each of the remaining teams follows a similar pattern, so we see that the total number of pairings is the sum: 15 + 14 + 13 + ... + 3 + 2 + 1. This is an evenly-spaced set, with an average of (15 + 1) / 2 = 8, and there are 15 terms in this set. Thus, the number of pairings is 8 x 15 = 120.

Answer: E