/sci/ - Science & Math

>>6819141
Here's what I find difficult:

Once m and n get big, my normal method of counting runs into partition problems.

For example, consider <span class="math">m=4[/spoiler] and <span class="math">n=6[/spoiler]. I effectively need to calculate the probability of the following outcomes, recorded as <span class="math">(X_1,X_2,X_3,X_4)[/spoiler]:

(6,0,0,0),(0,6,0,0),(0,0,6,0),(0,0,0,6)
(5,1,0,0)(5,0,1,0),(5,0,0,1),(1,5,0,0),(0,5,1,0),(0,5,0,1),(1,0,5,0),(0,1,5,0),(0,0,5,1),(1,0,0,5),(0,1,0,5),(0,0,1,5)
(4,2,0,0) x the same number of rearrangements
(4,1,1,0) x "
(3,1,1,1) x four arrangements
etc.
...

I can express it using partitions (with conditions) and multinomial coefficients, but I can't express it directly in terms of m and n. Is there some way to deal with the partitions in a clever way?