Question

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

(i) There is no restriction on the selection. 

(ii) A particular player is always chosen. 

(iii) A particular player is never chosen.

Answer 1

(i) Total number of ways of selecting 11 players out of 15 = ¹⁵C₁₁

= ¹⁵C_15-4

= ¹⁵C₄

\(= \frac{15 \times 14 \times 13 \times 12}{4\times 3 \times 2 \times 1}\)

\(= 1365\)

(ii) When a particular player is always chosen, then we will have to choose 10 players out of remaining 14.

∴ required number of ways = 14C10

= ¹⁴C_14-10

= ¹⁴C₄

\(= \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1}\)

\(= 1001\)

(iii) When a particular player is never chosen, then we will have to choose 11 players out of remaining 14.

∴ required number of ways = 14C11

= ¹⁴C_14-11

= ¹⁴C₃

\(= \frac{14 \times 13 \times 12}{ 3 \times 2 \times 1}\)

\(= 364\)

Answer 2

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

¹⁵C₁₁ = ¹⁵C₄

= 15 x 7 x 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

= 91 x 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 = \(\frac{14\times13\times12}{3\times2\times1}\) = 364 ways.