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In a football championship 153 matches were played. Every team played one match with each other team. How many teams participated in the championship?
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Verified Answer
The correct answer is:
18
Let total no. of team participated in a championship ben. Since, every team played one match with each other team.
$\therefore{ }^{n} C_{2}=153 \Rightarrow \frac{n !}{2 !(n-2) !}=153$
$\Rightarrow \frac{n(n-1)(n-2) !}{2 !(n-2) !}=153 \Rightarrow \frac{n(n-1)}{2}=153$
$\Rightarrow n(n-1)=306$
$\Rightarrow n^{2}-n-306=0$
$\Rightarrow n^{2}-18 n+17 n-306=0$
$\Rightarrow n(n-18)+17(n-18)=0$
$\Rightarrow n=18,-17$
$n$ cannot be negative $\therefore n \neq-17$
$\Rightarrow n=18
$\therefore{ }^{n} C_{2}=153 \Rightarrow \frac{n !}{2 !(n-2) !}=153$
$\Rightarrow \frac{n(n-1)(n-2) !}{2 !(n-2) !}=153 \Rightarrow \frac{n(n-1)}{2}=153$
$\Rightarrow n(n-1)=306$
$\Rightarrow n^{2}-n-306=0$
$\Rightarrow n^{2}-18 n+17 n-306=0$
$\Rightarrow n(n-18)+17(n-18)=0$
$\Rightarrow n=18,-17$
$n$ cannot be negative $\therefore n \neq-17$
$\Rightarrow n=18
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