1 Find the number of ways in which a mixed double tennis game can be arranged between 10 players consisting - Maths - Permutations and Combinations

Question

1 Find the number of ways in which a mixed double tennis game
can be arranged between 10 players consisting of 6 men and 4
women
2 In a hockey tournament , a total of 153 matches were played If
each team played one match with every other team, find the total
number of teams that participated in the tournament

Lalit Mehra · Accepted Answer

1 Number of ways to select two men out of six men $= {}^{6}{C_{2}}$ Number of ways to select two women out of four women $= {}^{4}{C_{2}}$ Number of ways of selecting the players for the mixed double $= {}^{6}{C_{2}} \times {}^{4}{C_{2}} = 15 \times 6 = 90$ Let M1, M2, W1 and W2 are selected players for the mixed double tennis game If M1 chooses W1, then M2 has W2 as the partner or if M1 chooses W2, then M2 has W1 as the partner âˆ´ There are two choices for the teams Thus, Number of ways in which mixed double tennis game be arranged = 90 Ã— 2 = 180 2 Let the number of teams participating in the tournament be n Number of matches played between the teams $= {}^{n}{C_{2}}$ $Given, {}^{n}{C_{2}} = 153 ∴ \frac{n (n - 1)}{2} = 153 \Rightarrow n^{2} - n - 306 = 0 \Rightarrow n^{2} - 18 n + 17 n - 306 = 0 \Rightarrow n (n - 18) + 17 (n - 18) = 0 \Rightarrow (n - 18) (n + 17) = 0 \Rightarrow n - 18 = 0 or n + 17 = 0 \Rightarrow n = 18 or n = - 17 ∴ n = 18 (∵ n c a n n o t b e n e g a t i v e)$ Thus, the number of teams participating in the tournament is 18