Question

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

Answer 1

To find the number of distinct sets of three marbles that can be selected from a bag of 6 green, 4 blue, and 2 red marbles, we consider the possible combinations and find that there are 10 distinct sets in total.

Step-by-step explanation:

The question asks for the number of distinct sets of three marbles that can be selected from a bag containing 6 green, 4 blue, and 2 red identical marbles. To determine this, we need to consider the different combinations in which we can draw 3 marbles. We will use a combinatorial approach to calculate the possible combinations, keeping in mind that the marbles of the same color are identical.

• First, consider drawing all three marbles of the same color, which gives us three possibilities: 3 green, 3 blue, or 3 red.
• Next, consider drawing two marbles of one color and one of another color. This yields several possibilities: green-green-blue, green-green-red, blue-blue-green, blue-blue-red, red-red-green, and red-red-blue.
• Lastly, consider one marble of each color, which yields just one possibility: green-blue-red.

Now we'll count the different sets:

• Three identical marbles: 3 possibilities (GGG, BBB, RRR).
• Two identical marbles and one different: 6 possibilities (GGB, GGR, BBG, BBR, RRG, RRB).
• One of each color: 1 possibility (GBR).

Adding them up, we get 3 + 6 + 1 = 10 distinct sets of marbles.

The correct answer is b) 10.