Probability word problem

Question

An urn contains  red,  white and  blue marbles. A child selects two marbles at random and without replacement from the urn.

Find the probability that the colors of the selected marbles are different.

Answer 1

p (both red) = (11/44) x (10/43)

p (both blue) = (21/44) x (20/43)

p (both white) = (12/44) x (11/43)

Adding these up gives 662/1892

If any other outcome happens, the colors must be different

So probability = 1 - (662/1892) = 1230 / 1892 = 615 / 946

Answer 2

This is 1 - P(both are the same color). As there are 11 red, 12 white, and 21 blue marbles, there are

C(11,2) combinations with 2 reds, C(12,2) combinations with 12 white, and C(21, 2) combinations with 2 blue, while there are C(11+12+21,2) = C(44, 2) possible combinations, so the probability both are the same color is

(C(11,2)+C(12,2)+C(21,2))/C(44,2) = (55+66+210)/946 = 331/946 and

P(both are not the same color) = 1 - 331/946 = 615/946