When a problem is being voted on in the Perplexus queue, the journeymen and scholars post comments and vote on the problem. A "thumbs up" (TU) scores +1 point, a "thumbs down" (TD) scores -1, and a comment with no vote scores 0.

Part one of this problem, which deals with combinations of TU and TD votes can be found here.

Suppose a problem in the queue has A responses and a score of B. Find a formula that gives the number of possible permutations of TU's, TD's and nonvoting comments the problem has received.

Note: For example a problem with 3 responses and a score of +1 has six possibilities:
{+1,0,0}
{0,+1,0}
{0,0,+1}
{+1,+1,-1}
{+1,-1,+1}
{-1,+1,+1}

(In reply to solution by Charlie)

It's apparent from the table that N(A,B) = N(A-1,B-1) + N(A-1,B) + N(A-1,B+1). I don't know if that will be helpful in finding a closed formula that doesn't use Sigma notation.

Posted by Charlie on 2007-05-23 15:46:24