In how many ways can a football team of 11 players be selected from 16 players?
Question:

In how many ways can a football team of 11 players be selected from 16 players? How many of these will

(i) include 2 particular players?

(ii) exclude 2 particular players?

Solution:

Number of ways in which 11 players can be selected out of $16={ }^{16} C_{11}=\frac{16 !}{11 ! 5 !}=\frac{16 \times 15 \times 14 \times 13 \times 12}{5 \times 4 \times 3 \times 2 \times 1}=4368$

(i) If 2 particular players are included, it would mean that out of 14 players, 9 players are selected.

Required number of ways $={ }^{14} C_{9}=\frac{14 !}{9 ! 5 !}=\frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1}=2002$

(ii) If 2 particular players are excluded, it would mean that out of 14 players, 11 players are selected.

Required number of ways $={ }^{14} C_{11}=\frac{14 !}{11 ! 3 !}=\frac{14 \times 13 \times 12}{3 \times 2 \times 1}=364$