How many possible orderings are there of the letters: N, N, B, B?

1 Answer
MathFact-orials.blogspot.com
Mar 13, 2017

6

Explanation:

When we're talking about doing orderings of letters, we're referring to a permutation, the general formula of which is:

#P_(n,k)=(n!)/((n-k)!); n="population", k="picks"#

Now if these 4 letters were all different, say #A,B, C, D#, we'd have:

#P_(4,4)=(4!)/((4-4)!)=(4!)/(0!)=(4!)/1=4! = 24#

But in our case we have 2 sets of 2 two letters that are identical. To get rid of the number of permutations that are the same (for instance, #N_1N_2B_1B_2# is the same as #N_2N_1B_2B_1#), we divide by the number of letters, factorial, that are the same. So for instance, we divide by #2!# to account for there being two N's. We also divide by #2!# again to account for there being two B's. And so that looks like:

#P_(4,4)/(2!2!)=(4!)/((4-4)!2!2!)=(4!)/(0!2!2!)=(4xx3!)/4=3! = 6#

With 6 permutations, we should be able to list them all:

NNBB
NBNB
NBBN
BBNN
BNBN
BNNB

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