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I form a multiset of symbols of size $n$:

ASDFJLAWEN

There are an infinite number of distinct symbols to choose from. Because it is a multiset, order does not matter. All that matters is the number of "distinct" symbols.

So JCKL is the same as ENOH because in each case there are four distinct symbols. AABJ is the same as KBBO because in each case there are three distinct symbols, two of which are the same.

ABAB is not the same as QQQL.

What is the number of such multisets?

Jbag1212
  • 1,698
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    This is equivalent to the problem of counting partitions of $n$. This is quite hard! – Kandinskij May 20 '24 at 16:49
  • @Kandinskij Oh yeah, you’re right. Is there a name for the number of partitions of size $n$ into $k$ distinct groups? For example 4,2,2 …4,3,1 … 3,3,2 are all partitions of 8 into 3 groups. So we know the number is at least 3 – Jbag1212 May 20 '24 at 16:55
  • Have a look at this question for a recursive formula: https://math.stackexchange.com/a/1908978/435819 – Paul Aljabar May 20 '24 at 19:08

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