Question

Manuel is putting 12 colored light bulbs into a string of lights. There are 3 red light bulbs, 6 pink light bulbs, and 3 orange light bulbs. How many distinct orders of light bulbs are there if two light bulbs of the same color are considered identical (not distinct)?

Answer 1

The total number of distinct orders of the light bulbs is 4,320.

Step-by-step explanation:

The number of distinct orders of light bulbs can be found by using the concept of permutations. Since we are considering two light bulbs of the same color to be identical, we can treat the red light bulbs as one group, the pink light bulbs as another group, and the orange light bulbs as the third group.

The number of distinct orders can be found by multiplying the number of ways to arrange each group. The number of ways to arrange the red light bulbs is 1, since they are all the same. The number of ways to arrange the pink light bulbs is 6!, which is the number of permutations of 6 objects. The number of ways to arrange the orange light bulbs is 3!, which is the number of permutations of 3 objects.

Therefore, the total number of distinct orders is 1 x 6! x 3! = 1 x 720 x 6 = 4,320.