In a bag are 6 identical green marbles, 4 identical blue marbles, and 2 identical red marbles. If 3 marbles are picked

[Gmat_mission]

In a bag are 6 identical green marbles, 4 identical blue marbles, and 2 identical red marbles. If 3 marbles are picked at random from the twelve in the bag, how many distinct sets of three can be selected?
a) 9
b) 10
c) 15
D) 21
E) 91

[spoiler]OA=A[/spoiler]

Source: Magoosh

[deloitte247]

Identical green marbles = 6
Identical red marbles = 2
Identical blue marbles = 4
Total marbles = 12

How many distinct sets of three can be selected?

Possible sets are :
- The selected 3 are different colors, 1 red, 1 green, and 1 blue -> only one way
- The selected 3 have 2 identical marbles and 1 different marble. I.e 2 red and 1 green, 2 blue and 1 red, 2 green and 1 blue
For all the listed ways, there are alternative combinations. i.e 2 red and 1 green or 2 green and 1 red, 2 blue and 1 red or 2 red and 1 blue e.t.c number of possible ways in this set = (3C2)*2
$$3C2=\frac{3!}{2!\left(3-2\right)!}=\frac{3\cdot2\cdot1}{2\cdot1\cdot1}=\frac{6}{2}=3$$
$$3\cdot2=6\ possible\ ways$$
- The selected 3 have the same color but since there are only 2 identical red marbles, it is only possible to select 3 blue or 3 green marbles. So, there are 2 possible ways to select from this set from the 3 possible sets above. Therefore, the total possible distinct sets of 3 that can be selected = 1 + 6 + 2 = 9

Answer = A

[Scott@TargetTestPrep]

In a bag are 6 identical green marbles, 4 identical blue marbles, and 2 identical red marbles. If 3 marbles are picked at random from the twelve in the bag, how many distinct sets of three can be selected?
a) 9
b) 10
c) 15
D) 21
E) 91

[spoiler]OA=A[/spoiler]

Source: Magoosh

There are so few ways to make the selections that it is easiest to simply list all the possibilities:

All 3 different colors: R, G, B - 1 way

2 different colors: RRG, RRB, BBR, BBG, GGR, GGB - 6 ways

All 3 the same color: BBB, GGG - 2 ways (note that we can’t have RRR because there are only 2 red marbles)

Thus, the total number of ways is 1 + 6 + 2 = 9.

Answer: A