13 red, 13 blue, 13 yellow, and 13 green. Marbles of each color are letters A through M. 5 marbles are chosen?

There are a total of 52 marbles, so the number of 5 marble choices is

(52 choose 5) = 2598960.

There are 13 ways of choosing the letter that is repeated.

The two marbles with repeated letters are of different colors, so there are (4 choose 2) ways of choosing the colors of these two marbles.

There are (12 choose 3) ways of choosing the other 3 letters.

For the other 3 letters, there are 4 ways of choosing the color of the marble with the letter coming first alphabetically, 4 ways of choosing the color of the marble with the letter coming second alphabetically, and 4 ways of choosing the color of the marble with the letter coming last alphabetically.

Therefore, there are 13*(4 choose 2)*(12 choose 3)*(4^3) = 1,098,240 five marble choices that consist of exactly 2 with the same letter and 3 others, different letters.

This also means that the probability of exactly 2 with the same letter and 3 others, different letters, is 1,098,240/2,598,960 or about 0.4226 .

May Jesus richly bless you today!