First slide

Combinations

Question

In a football championship, there were played 153 matches. Every team played one match with each other. The number of team 8 participating in the championship is

Moderate

A

17
B

18
C

9
D

13

Solution

Let there are n teams.
Each team play to every other team in $^{n} C_{2}$ , ways
$\begin{array}{ll} ∴^{n} C_{2} = 153 \\ \Rightarrow \frac{n!}{(n - 2)! 2!} = 153 \end{array}$
$\begin{array}{lr} \Rightarrow n (n - 1) = 306 \\ \Rightarrow n^{2} - n - 306 = 0 \Rightarrow (n - 18) (n + 17) = 0 \\ \Rightarrow n = 18 [∵ n is never negative] \end{array}$

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