How is locating 21.5 on a number line the same as locating 1.5 on a number line? How is it different?
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There are a total of 2 + 3 + 6 = 11 letters, which gives rise to 11! possible permutations.
Consider one such permutation:
BBHHHOOOOOO
I write one of the Bs in the bold for emphasis. This permutation is not distinguishable (but for the temporary boldface) from
BBHHHOOOOOO
which is to say they both count as the same permutation. To avoid counting this twice, you would divide the total number of permutations by the number of ways you can permute the identical character. In the case of Bs, this can be done in 2! = 2 ways.
Similarly, the 3 Hs can be rearranged in 3! = 6 ways, and the 6 Os can be rearranged in 6! = 720 ways.
Then the total number of distinguishable permutations is
11! / (2! • 3! • 6!) = 4620
As a formula: given a word of length with
different characters that respectively occur
times, the total number of distinct permutations is
where .
Try this with some simple examples:
• GREECE
This has length 6, with E occurring three times, and every other letter occurs once. The total number of distinct permutations of GREECE is
6! / (1! • 1! • 1! • 3!) = 120
• BANANA
This also has length 6, with B occurring once, N twice, and A three times, so the total number of distinct permutations of BANANA is
6! / (1! • 2! • 3!) = 60
• STATISTICALLY
This has length 13, with one each of C and Y; two each of S, A, I, L; and three Ts. Then the number of distinct permutations is
13! / (1! • 1! • 2! • 2! • 2! • 2! • 3!) = 64,864,800
