Permuations and Combinations

Question

Find the number of ways in which a team of 6 batsmen, 4 bowlers and a wicket keeper may be selected from a squad of 8 batsmen, 6 bowlers and  2 wicket keepers.

 

8C6x6C4x2C1=840

 

Find the number of ways in which 

(a) this team may be selected if it is to include 4 specified batsmen and 2 specified bowlers Ans 72

(b) the batsmen may be selected from the 8 available given that 2 particular batsmen cannot be selected together. Ans 12

Answer 1

For a, we have to choose 2 additional batsmen from the remaining 4 and 2 additional bowlers from the remaining 4.  So, we have:

 

4C2x4C2x2C1 = 6x6x2 = 72

 

For b, the key is the batsmen.  There are two cases.  If one of the special 2 is chosen, then we can consider the batsmen as two categories, so we have 6C5x2C1 = 12 possibilities.  If neither of the special two are picked (which appears by the wording to be an option), then we have 6C6 = 1 possibility.  Adding these, we have 13 ways to select the batsmen, as Daniel pointed out.

 

Incidentally, you could also get this answer by taking the total (8C6=28) minus the possibilities that include both of these players (6C4=15) to get 28-15=13 possibilities that don't include both of these players.