How many 4 digits odd numbers can be formed?

Quite appropriately, the word hippopotomonstrosesquipedalianism is used to describe words that are enormously long. In how many ways can its letters be arranged?

In the $33$ letters of the word:

• there are 5 occurrences of o
• p, i and s both occur 4 times each, and
• t, m, n, e, and a all occur 2 times each

resulting in $\displaystyle{\frac{33!}{5! (4!)^3 2^5}}$ possible ways to arrange the letters.

For the curious, the above expression evaluates to $163576434454494269015808000000$.

R
> counts = c(1,4,4,5,2,2,2,4,1,2,1,1,1,2,1)
          #  h i p o t m n s r e q u d a l
> factorial(sum(counts))/prod(factorial(counts))

There are five odd digits {1,3,5,7,9} so our numbers will be made using only these digits. This is the same number of possibilities as if you had a base 5 number system where the allowed digits are {0,1,2,3,4}. For a 1 digit base-5, there are 5 possibilities, for a two digit base-5 there are 5² = 25 possibilities, so for a 4-digit, there are 5^4 = 625possible. This is the same number you will have with only using odd digits.

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