Answer to Question #8891 in Statistics and Probability for Rahman

Question

An urn contains 7 white balls and 10 red balls. Five balls are selected. In how many ways can the 5 balls be drawn from the total of 17 balls:
iii) If all 5 balls are red?
iv) If all 5 balls are white?
v) If at least 4 are red balls?
vi) If at least 2 are of same color?

Answer 1

An urn contains 7 white balls and 10 red balls. Five balls are selected. In how many ways can the 5 balls be drawn from the total of 17 balls:
iii) If all 5 balls are red?

There are C(10,5) = 252 ways to drawn 5 red balls from 10 red balls.

iv) If all 5 balls are white?

There are C(7,5) = 21 ways to drawn 5 white balls from 7 white balls.

v) If at least 4 are red balls?

There are C(10,4) = 210 ways to drawn 4 red balls from 10 red balls and C(17-4,1) = 13 ways to drawn one other (red or white) ball. So, totally 210+13 = 223 ways.

vi) If at least 2 are of same color?

There are C(10,2) = 45 ways to drawn 2 red balls and C(7,2) = 21 ways to drawn 2 white balls; so, there are 45+21 = 66 ways to drawn 2 balls of same color. At last, there are C(17-2,3) = 455 ways to drawn three other balls, so the total is 66+455 = 521.