Re: Permutations and combinations doubt

Posted by Amit Goyal on Apr 01, 2014; 3:28pm
URL: http://discussion-forum.276.s1.nabble.com/Permutations-and-combinations-doubt-tp7585896p7585916.html

Q. Find number of ways of distributing 5 distinct balls to 3 distinct boxes so that no box is empty.
A.

Number of ways of distributing 5 distinct balls to 3 distinct boxes
= 3 x 3 x 3 x 3 x 3

Number of ways of distributing 5 distinct balls to 2 specific boxes (say, Box 1 and Box 2)
= 2 x 2 x 2 x 2 x 2

Number of ways of distributing 5 distinct balls to 1 specific box (say, Box 1)
= 1 x 1 x 1 x 1 x 1

Number of ways of distributing 5 distinct balls to exactly two boxes so that none of those two box is empty
= 3[(2 x 2 x 2 x 2 x 2) - 2(1 x 1 x 1 x 1 x 1)]

Number of ways of distributing 5 distinct balls to exactly 1 box
= 3[1 x 1 x 1 x 1 x 1]

Number of ways of distributing 5 distinct balls to 3 distinct boxes so that no box is empty
= Number of ways of distributing 5 distinct balls to 3 distinct boxes - Number of ways of distributing 5 distinct balls to exactly two boxes so that none of those two box is empty - Number of ways of distributing 5 distinct balls to exactly 1 box
= [3 x 3 x 3 x 3 x 3] - 3[(2 x 2 x 2 x 2 x 2) - 2(1 x 1 x 1 x 1 x 1)] -  3[1 x 1 x 1 x 1 x 1]
= 243 - 90 - 3
= 150