Simple probability question on combinatorics

[kougou]

Homework Statement

I am trying to understand the question:

An urn contains n red and m blue balls. They are withdrawn one at a time until a total of r(r≤n) red balls have been withdrawn. Find the probability that a total of k balls are withdrawn.

The solution is given as,
Sample Space:
(n+mk−1)×(n+m−k+1),
Event(kth ball is rth red ball)=(n)C(r−1)×(m)C(n−r)×(n−r+1)


I also don't understand why the solution use (n)C(r-1)x (m)C(n-r);
I think the ordering of drawing the red balls and the blue balls are important, such that RBBR
is different from BBRR. But the solution says (n)C(r-1), which means choosing (r-1) red balls from n red balls, and choose the remaining (n-r) from the blue balls.

This as if saying that they want the combination of R1 B1 B2 R2 and that's different from
R3 B3 B4 R5
Any ideas?

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

hi kougou!

An urn contains n red and m blue balls. They are withdrawn one at a time until a total of r(r≤n) red balls have been withdrawn. Find the probability that a total of k balls are withdrawn.

i'm not sure i understand what you're asking

the question is asking for the number k such that the first k-1 balls contain exactly r-1 reds, and the kth ball is red (out of the remaining n-r+1 reds and … blues)