A badminton club has 21 members. What is the ratio of number of 6-memb

A 6-member committee can be formed by selecting any 6 persons from the 21 members of the club.
Number of ways of selecting any 6 persons from 21 persons = \(21_C_6\) = 21 * 20 * 19 * 18 * 17 * 16 / 6 * 5 * 4 * 3 * 2 * 1

A 5-member committee can be formed by selecting any 5 persons from the 21 members of the club.
Number of ways of selecting any 5 persons from 21 persons = \(21_C_5\) = 21 * 20 * 19 * 18 * 17 / 5 * 4 * 3 * 2 * 1
Therefore, ratio of number of 6-member committees to ratio of number of 5-member committees = ((21 * 20 * 19 * 18 * 17 * 16)/( 6 * 5 * 4 * 3 * 2 * 1))/((21 * 20 * 19 * 18 * 17)/(5 * 4 * 3 * 2 * 1))
Simplifying, the required ratio = (21 * 20 * 19 * 18 * 17 * 16)/(6 * 5 * 4 * 3 * 2 * 1) * (5 * 4 * 3 * 2 * 1)/(21 * 20 * 19 * 18 * 17)

= \(\frac{16 }{ 6}\) = \(\frac{8 }{ 3}\)

The correct answer option is C.