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How many different three-digit numbers can I make with three 3s, two 2s, and one 1? I'd like to use actual calculations instead of just counting them all.

What I am curious about is how am I supposed to do this kind of problem when the numbers are more complicated? For example, "How many different five-digit numbers are available with x threes, y twos, and z ones?"

user10219414
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  • If you don't say what have you attempted to do, our impression (maybe false) is that you ask us to do your homework. – Jean Marie May 03 '21 at 22:11
  • What have you tried and what do you know? Are you familiar with any of the Combinatorial_principles? (Also see How to find number of numbers formed with given digits?.) – Vepir May 03 '21 at 22:16
  • For this particular problem, my personal attack would be leveraging that there are three $3$'s available. So I would break it up into 4 cases, depending on how many of the $3$'s were used. Note that there are $(3!)$ ways of permuting the elements ${1,2,3},$ while only $(3)$ ways of permuting the elements ${3,2,2}.$ – user2661923 May 03 '21 at 22:34
  • The way my teacher told me to do this problem is by counting all of the possibilities one by one. (122, 123, 132, 133, 212, 213, 221, 223, 231, 232, 233, 312, 313, 321, 322, 323, 331, 332, 333) – user10219414 May 04 '21 at 00:58
  • What I am curious about is how am I supposed to do this kind of problem when the numbers are more complicated? For example, "How many different five-digit numbers are available with x threes, y twos, and z ones?" – user10219414 May 04 '21 at 01:05
  • Perhaps OEIS sequence A172528 can help you. – Somos May 04 '21 at 02:06
  • Your spirit of providing context is appreciated, but it's better to put these details in the original post, rather than as comments underneath. (Comments can get deleted.) – Brian Tung May 04 '21 at 03:16
  • Exponential generating functions provide a general method for solving this kind of question. There are many resources for learning about generating functions here: https://math.stackexchange.com/questions/3142386/how-can-i-learn-about-generating-functions/3142387 – awkward May 04 '21 at 13:09
  • @awkward I agree that exponential generating functions are the way to go. However, I think that this begs the question of the problem composer's intent for the OP (i.e. original poster). Based on "the way my teacher told me is by counting...", I think the intent is only for the OP to manually count. Then, your generating functions idea, which makes sense, answers the OP's question re: "What I am curious about is how...". Perhaps what the OP doesn't realize, is that the topic of generating functions, by itself, will take significant time to study. – user2661923 May 04 '21 at 16:57

1 Answers1

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After analyzing some data using this pattern, I have discovered that if the amount of different numbers (example: four 4s, three 3s, two 2s, and one 1 would be 5 different numbers) would be n, you would use this equation (n is a positive integer greater than or equal to the amount of digits(3 in this equation)):

p = (n - 1)(n - 1) + ((n - 1)(n(n) - 2) + (n - 2))

(p means the possibilities)

Tell me if you found a simpler way of writing this.

gbe
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