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Veseljchak [2.6K]
2 years ago
7

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.
Mathematics
2 answers:
Nina [5.8K]2 years ago
5 0

Answer:

<em>5</em><em> players participated in the tournament.</em>

Step-by-step 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\times ^nC_2

But it is given to be 20, so

\Rightarrow \ 2\times ^nC_2=20

\Rightarrow \ ^nC_2=10

\Rightarrow \dfrac{n!}{2!(n-2)!}=10

\Rightarrow \dfrac{n(n-1)}{2}=10

\Rightarrow {n(n-1)=20

As 5\times 4=20, so we get n=5.

Therefore, 5 players participated in the tournament.

Monica [59]2 years ago
5 0

Answer:


Step-by-step 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 = \frac{20}{2}

⇒^nC_2 =10  --(a)

Formula of combination:

⇒\frac{n!}{r! * (n-r)!}

So, solving (a) further using formula

⇒\frac{n!}{2! * (n-2)!}=10

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

⇒\frac{n*(n-1)}{2*1}=10

⇒\frac{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.






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