Question

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

Answer 1

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