Question

Fifteen people took part in a chess competition. Every player plays with other players once. Find the number of games played in the tournament

Answer 1

Total number of people available for chess competition $=15$
Number of people needed to play a chess game $=2$

We need to choose $2$ different people from $15$ people for each chess game.

The number of games played would be ${}^{15}{C}_2$.

Here, $n=15$ and $r=2$.

${}^{15}{C}_2=\frac{15!}{2!(15-2)!}$ ($\because {}^n{C}_r=\frac{n!}{r!(n-r)!}$)

${}^{15}{C}_2=\frac{15!}{2!13!}$

${}^{15}{C}_2=\frac{15\times 14\times 13!}{2!13!}$

${}^{15}{C}_2=\frac{15\times 14}{2\times 1}$

${}^{15}{C}_2=15\times 7$
${}^{15}{C}_2=105$

The number of games played if every person must play each of the others is $\text{105}$.