Question

In a small chess tournament, 20 matches were played. Find out how many people were involved if it is known that each participant played 2 games with every other participant in the tournament.

Answer 1

Explanation:

Given :

20 matches were played in a small chess tournament.

Each participant played 2 games with every other participant in the tournament.

To Find : how many people were involved?

Solution :

Let no. of players involved be n

Since we know that for every match there should be two players out of n

So, number of ways they can play :

`^nC_2`

We are also given that each participant played 2 games with every other participant.

So, total no. of games played =
`2 * ^nC_2`

Since we are given that total no. games played = 20

⇒
`2 * ^nC_2 = 20`

⇒
`^nC_2 = (20)/(2)`

⇒
`^nC_2 =10` --(a)

Formula of combination:

⇒
`(n!)/(r! * (n-r)!)`

So, solving (a) further using formula

⇒
`(n!)/(2! * (n-2)!)=10`

⇒
`(n*(n-1)*(n-2)!)/(2! * (n-2)!)=10`

⇒
`(n*(n-1))/(2*1)=10`

⇒
`(n^(2) -n)/(2*1)=10`

⇒
`n^(2) -n=10*2`

⇒
`n^(2) -n=20`

⇒
`n^(2) -n-20=0`

⇒
`n^(2) -5n+4n-20=0`

⇒
`n(n-5)+4(n-5)=0`

⇒
`(n-5)(n+4)=0`

⇒
`(n-5) =0 , (n+4)=0`

⇒ n = 5 , n =-4

Neglect the negative value since no. of players cannot be negative.

Thus no. of player involved is 5.

Answer 2

5 players participated in the tournament.

Explanation:

In a small chess tournament, 20 matches were played.

Let us assume that n number of players participated in the tournament

As in each game 2 players play, so the number of ways they can play is,

`=\ ^nC_2`

As they played 2 games with every other participant in the tournament.

So the total number of games is,

`=\ 2* ^nC_2`

But it is given to be 20, so

`\Rightarrow \ 2* ^nC_2=20`

`\Rightarrow \ ^nC_2=10`

`\Rightarrow (n!)/(2!(n-2)!)=10`

`\Rightarrow (n(n-1))/(2)=10`

`\Rightarrow {n(n-1)=20`

As
`5* 4=20`, so we get n=5.

Therefore, 5 players participated in the tournament.