1. The number of selections when at most 1 all-rounder and 1 wicket keeper will play is

Question

If a cricket team of 11 players is to be selected from 8 batsmen, 6 bowlers, 4 all-rounder and 2 wicket keepers, then

1. The number of selections when at most 1 all-rounder and 1 wicket keeper will play is

(A)  ⁴C₁ ⋅ ¹⁴C₁₀ +  ²C₁ ⋅ ¹⁴C₁₀ +  ⁴C₁ ⋅ ²C₁ ²C₁ ⋅ ¹⁴C₉ + ¹⁴C₁₁ 

(B)  ⁴C₁ ⋅ ¹⁵C₁₁ + ¹⁵C₁₁ 

2. Number of selection when 2 particular batsmen do not want to play when a particular bowler will play is 

(A)  ¹⁷C₁₀ + ¹⁹C₁₁ 

(B)  ¹⁷C₁₀ + ¹⁹C₁₁ + ¹⁷C₁₁ 

(C) ¹⁷C₁₀ + ²⁰C₁₁ 

(D) ¹⁹C₁₀ + ¹⁹C₁₁

3.  The number of selections when a particular batsman and a particular wicket keeper do not want to play together is

(A)  2⋅ ¹⁸C₁₀

(B)  ¹⁹C₁₁ + ¹⁸C₁₀ 

(C)  ¹⁹C₁₀ + ¹⁹C₁₁ 

(D)  None of these 

Answer 1

Correct option 1. (A) 2. (B) 3. (B)

1.  When 1 all-rounder and 10 players from bowlers and batsman play number of ways = ⁴C₁ ⋅ ¹⁴C₁₀ 

When 1 wicket keeper and 10 players from bowlers and batsman play number of ways = ²C₁ ⋅ ¹⁴C₁₀ 

When 1 all-rounder 1 wicket keeper and 9 from batsmen and bowlers play number of ways = ⁴C₁ ⋅ ²C₁ ⋅ ¹⁴C⁹

When all 11 players play from bowlers and batsmen then the number of ways = ¹⁴C₁₁ 

Therefore, the total number of selections

= ⁴C₁ ⋅ ¹⁴C₁₀ + ²C₁ ⋅ ¹⁴C₁₀ + ⁴C₁ ⋅ ²C₁₀ ⋅ ¹⁴C₉ + ¹⁴C₁₁

2. If 2 batsmen do not want to play then the rest of 10 players can be selected from 17 other players, number of selection = ¹⁷C₁₀.

If the particular bowler does not play then number of selection = ¹⁹C₁₁ .

If all the three do not play, number of selection = ¹⁷C₁₀ .

Therefore, the total number of selections = ¹⁷C₁₀ + ¹⁹C₁₁ +

¹⁷C₁₁.

3.  If the particular batsman is selected then the rest of 10 players can be selected in ¹⁸C₁₀ ways.

If particular wicket keeper is selected then rest of 10 players can be selected in ¹⁸C₁₀ ways. 

If both are not selected the number of ways  = ¹⁸C₁₁

Therefore the total number of ways =   2⋅ ¹⁸C₁₀ + ¹⁸C₁₁

= ¹⁹C₁₁ + ¹⁸C₁₀