Seven sportsmen (named A,B,C,D,E,F and G) buy 7 hats choosing randomly out of 4 available colors (say G,R,W and Y).
What is the probability that all the colors were chosen?

Total Number of ways in which caps can be selected

= 4^7 (A can select 4 ways, B can select 4 ways , ....)

Let G,R,W,Y are already selected, the other caps can be selected as

1) 3 of different colors

2) 2 of same color, 1 of other color

3) 3 of the same color

Case 1)

No. of ways 3 colors can be selected = 4C3 = 4

No. of ways they can be selected by 7 players = 7!*4/(2!*2!*2!)  = 2520

Case 2)

No. of ways 2 same 1 different can be selected = 4C2  = 6

No. of ways they can be selected by 7 players = 12*7!/(3!*2!) = 5040

 

Case 3)

No. of ways 3 same can be selected = 4

No. of ways they can be selected by 7 players = 4*7!/4! = 840

Total No. of ways = 8400

=> Probability = 8400/16384 = 0.5126953125

Edited on May 10, 2010, 9:35 am

Posted by Praneeth on 2010-05-10 09:19:54