Question

Q194. In a chess tournament each of six players will play every other player exactly once.
How many matches will be played during the tournament?
a) 12
b) 15
c) 30
d) 36​

Answer 1

Topic

Permutation and Combination

Given

In a chess tournament, each of six players play with every other Player exactly once.

To Find

Total number of matches played during the tournament.

Solution

Technique 1

Every player will play exactly one match with other players.

As we know, a chess match can be played between two players at a time. So, we will select two players out of total six players and permute them.

Applying permutation,

$\binom{6}{2}$

$\dfrac{6!}{4!2!}$

$\dfrac{4!.5.6}{4!2!}$

$\dfrac{5.6}{2}$

$\dfrac{30}{2}$

15

So, total 15 matches will be played.

Technique 2

We will count total number of matches manually but smartly.

Name players as following :

Player 1

Player 2

Player 3

Player 4

Player 5

Player 6

Player 6 will play matches with all 5 other players.

Player 5 will play matches with all 4 other players as he have already played with Player 6.

Player 4 will play matches with all 3 other players as he have already played with Player 5 and 6.

Player 3 will play matches with all 2 other players as he have already played with Player 4,5 and 6.

Player 2 will play matches with Player 1 only as he have already played with all other players.

Now, summing up for total number of matches.

6 + 5 + 4 + 3 + 2 + 1

15

Answer

So, total 15 matches were played during the tournament.