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

Arrangements, combinations and permutations, probability, ...
rdybowski
Posts: 2
Joined: Sun Jun 05, 2016 10:52 pm

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

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
Posts: 2
Joined: Sun Jun 05, 2016 10:52 pm

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

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

MuthuVeerappanR
Posts: 423
Joined: Sun Mar 22, 2015 2:30 pm
Location: India
Contact:

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

It would be great if you can share the solution here... It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.