How many ways can you arrange letters in a word?

appetite

• Starting Point: There are 8! ways to arrange the letters of the word appetite.
  • That's 8! permutations of the letters.
  • 8! = 40,320.
• Now we account for the swapability in the letter piles:
  • There are 2 e's, 2 p's, 2 t's, 1 a, and 1 i.
  • So we reduce 8! = 40,320 by 2!2!2!1!1! = 8.
  • 40,320/8 = 5040.
• There are 5040 ways to arrange the letters of appetite.

assassin

• Starting Point: There are 8! ways to arrange the letters of the word assassin.
  • That's 8! permutations of the letters.
  • 8! = 40,320.
• Now we account for the swapability in the letter piles:
  • There are 4 s's, 2 a's, 1 i, and 1 n.
  • So we reduce 8! = 40,320 by 4!2!1!1! = 48.
  • 40,320/48 = 840.
• There are 840 ways to arrange the letters of assassin.

mammal

• Starting Point: There are 6! ways to arrange the letters of the word mammal.
  • That's 6! permutations of the letters.
  • 6! = 720.
• Now we account for the swapability in the letter piles:
  • There are 3 m's, 2 a's, and 1 l.
  • So we reduce 6! = 720 by 3!2!1! = 12.
  • 720/12 = 60.
• There are 60 ways to arrange the letters of mammal.

A Final Note

We accounted for the swapability of letters within letter piles (intrapile swapability). But we did not account for swapping the piles themselves like we do in poker hands (interpile swapability).

That's because one letter pile (say, 2 a's) cannot substitute for another letter pile, even of the same size (say, 2 p's).

The piles in our poker-hand problems haven't been assigned a rank yet, so they're just generic triples, pairs, singles, etc. If we assigned ranks to them, we couldn't swap them (e.g., a pair of Jacks couldn't swap with a pair of nines).

So in that case, the poker card count would lose its interpile swapability, just as in the arranging-letters-of-a-word problem.