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

Answer:

5 people

Step-by-step explanation:

Answer 2

Answer:

5

Step-by-step explanation:

We are given that 20 matches were played in a small chess tournament.

We are also given that Each participant played 2 games with every other participant in the tournament.

We are required to find out how many people were involved in the game.

So, First Let no. of players involved be n

Since we are given  for every match there should be two players out of n

Thus, number of ways they can play  a match : $^nC_2$

Since we know that each participant played 2 games with every other participant.

Thus , The total no. of games played  =$2 * ^nC_2$

We can see that 20 matches were played in total

⇒$2 * ^nC_2 = 20$

⇒$^nC_2=\frac{20}{2}$

⇒$^nC_2=10$

Thus using the combination formula i.e. $\frac{n!}{r! * (n-r)!}$

Since total players = n and r = 2

⇒ $\frac{n!}{2! * (n-2)!}=10$

⇒ $\frac{n*(n-1)*(n-2)!}{2! * (n-2)!}=10$

⇒ $\frac{n*(n-1)}{2*1}=10$

⇒ $n*(n-1)=10*2$

⇒ $n*(n-1)=20$

⇒ $n*(n-1)-20=0$

⇒ $n^{2}-n-20=0$

⇒ $n^{2}-5n+4n-20=0$

⇒ $n(n-5)+4(n-25)=0$

⇒(n+4)=0  ,  (n-5)=0

⇒ n = -4, 5

Number of players cannot be negative so neglect n = -4

Thus , Number of players involved were 5