Question

In a football championship, $153$ matches were played. Every two teams played one match with each other. The number of teams participating in the championship is

Answer 1

Let the number of teams participating in the
championship be $n$.

Total match played among teams such that every two teams played one match with each other is equal to number of ways of selecting $2$ objects from given n distinct objects = ${}^n{C}_2$

${}^n{C}_2=153$,

$(\because \text{Given that 153 matches were played)}$

$\Rightarrow \frac{n!}{(n-2)!2!}=153[\because {}^n{C}_r=\frac{n!}{(n-r)!r!}]$
$$\begin{array}{c}\Rightarrow \frac{n(n-1)(n-2)!}{(n-2)!2!}=153\\ \Rightarrow n^2-n=153\times 2=306\\ \Rightarrow n^2-n-306=0\dots \left(1\right)\\ n^2-18n+17n-306=0\\ \Rightarrow n(n-18)+17(n-18)=0\\ \Rightarrow (n-18)(n+17)=0\\ \Rightarrow n-18=0\text{or}n+17=0\\ \Rightarrow n=18\text{or}n=-17\left(\text{which is not possible}\right)\end{array}$$
Hence, the number of teams participating in the championship is $18.$