Question

In how many ways can a cricket team of 11 players be chosen out of a batch of 15 players?

1. There is no restriction on the selection.
2. A particular player is always chosen.
3. A particular player is never chosen.

Answer 1

(i) Number of ways choosing 11 players from 15 is ¹⁵C₁₁ = ¹⁵C₄ 

`= (15 xx 14 xx 13 xx 12)/(4 xx 3 xx 2 xx 1)`

= 15 × 7 × 13

= 1365.

(ii) If a particular is always chosen there will be only 14 players left put, in which 10 are to selected in ¹⁴C₁₀ ways.

¹⁴C₁₀ = ¹⁴C₄ 

`= (14xx13xx12xx11)/(4xx3xx2xx1)`

`= (14 xx 13 xx 11)/2`

= 91 × 11

= 1001 ways

(iii) If a particular player is never chosen we have to select 11 players out of remaining 14 players in ¹⁴C₁₁ ways.

i.e., ¹⁴C₃ ways = `(14 xx 13 xx 12)/(3xx2xx1)` = 364 ways.