Re: Permutations and combinations doubt

Posted by Amit Goyal on Apr 02, 2014; 1:58pm
URL: http://discussion-forum.276.s1.nabble.com/Permutations-and-combinations-doubt-tp7585896p7585980.html

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