       Re: k-permutations enumeration

• To: mathgroup at smc.vnet.net
• Subject: [mg116326] Re: k-permutations enumeration
• From: Ray Koopman <koopman at sfu.ca>
• Date: Fri, 11 Feb 2011 04:14:28 -0500 (EST)
• References: <ij0eaq\$9eo\$1@smc.vnet.net>

```On Feb 10, 2:23 am, Michal Kvasnicka <michal.kvasni... at gmail.com>
wrote:
> How can I find the number of k-permutations of n objects, where there
> are x types of objects, and r1, r2, r3 ... rx give the number of each
> type of object?
>
> Example:
> I have 20 letters from the alphabet. There are some duplicates - 4 of
> them are a, 5 of them are b, 8 of them are c, and 3 are d. How many
> unique 15-letter permutations can I make?
>
> In the example:
>
> n = 20
> k = 15
> x = 4
> r1 = 4, r2 = 5, r3 = 8, r4 = 3
>
> Furthermore, if there isn't a straightforward solution: how
> efficiently can this problem be solved? Any Mathematica code?

Please disregard my previous post, which counted only
"words" containing all of the x different "letters".

k = 15; r = {4,5,8,3};
Tr[Multinomial@@@Select[Tuples[Range[0,#]&/@r],Tr@#==k&]]

187957770

```

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