Total number of ways of selecting n balls from an urn with k colours

Arrangements, combinations and permutations, probability, ...
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rdybowski
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Joined: Sun Jun 05, 2016 10:52 pm

Total number of ways of selecting n balls from an urn with k colours

Post by rdybowski » Sun Jun 05, 2016 10:58 pm

Suppose I have an urn containing balls D of k different colors. Let D_i be the subset of D of all the balls having the i-th colour. The probability of randomly selecting (without replacement) y_1 balls from D_1, y_2 balls from D_2, . . . , y_k balls from D_k is, of course, given by the multivariate hypergeometric distribution. What I am interested in is the total number of ways of selecting n balls from D.

For example, if D = [2 | 1 | 2] (i.e., k = 3), and n = 3, then the possible ways of selecting n balls from D are

[2 | 1 | 0]

[2 | 0 | 1]

[1 | 1 | 1]

[1 | 0 | 2]

[0 | 1 | 2]

In the above example, the number of ways of selecting 3 balls from D is 5, but is there a general mathematical expression for the total number of ways of selecting n balls from D?

rdybowski
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Joined: Sun Jun 05, 2016 10:52 pm

Re: Total number of ways of selecting n balls from an urn with k colours

Post by rdybowski » Wed Jun 08, 2016 11:14 am

[Solved]
The answer is given on page 138 of Charalambides, C. A. (2002) Enumerative Combinatorics, Chapman & Hall/CRC.

MuthuVeerappanR
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Re: Total number of ways of selecting n balls from an urn with k colours

Post by MuthuVeerappanR » Thu Jun 09, 2016 7:09 pm

It would be great if you can share the solution here...
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