First arrange the words in order:
AAABNN
so there are 3 A's, 1 B and 2 N's.
If there were no constraints,
Number of permutations = (3+1+2)!/(3!1!2!)=6!/(6*1*2)=720/(6*2)=60
Number of words that begin with B is to take out the B, that leaves us with
AAANN => 3 A's and 2 N's
=> 5!/(3!2!) = 120/(6*2)=10
Therefore the number of words that do NOT begin with B is
60-10 = 50 ways
