An urn has $25$ balls, of which $15$ are red and $10$ are white.

Question

An urn has $25$ balls, of which $15$ are red and $10$ are white. Select $5$ balls.
a. How many samples exist?
b. How many samples are all red?
c. How many samples have $3$ red?
d. How many samples have at least $4$ red?

I understand a... unfortunately, that's about it.

For a, I got $C(25,5) = 53130$.

Answer 1

(a) $C(25,5)$

(b) $C(15,5)$

(c) Three balls must be red which is equal to $C(15,3)$ and two balls must be white which is equal to $C(10,2)$. So, in total we have $C(15,3)C(10,2)$

(d) Here we must select samples with four red balls (and one white) and five red balls (and zero white). Apply the same rule as in (b) to compute the respective number of samples and sum them up: $C(15,4)C(10,1) + C(15,5)$

Answer 2

(5 red, 0 white) -- ${15\choose 5}$

(4 red, 1 white) -- ${15\choose 4}{10\choose1}$

(3 red, 2 white) -- ${15\choose 3}{10\choose2}$