Pablo is putting 12 colored light bulbs into a string of lights. There are 3 pink light bulbs, 6 blue light bulbs, and 3 red light bulbs. How many distinct orders of light bulbs are there if two light bulbs of the same color are considered identical (not distinct)?
Question
Pablo is putting 12 colored light bulbs into a string of lights. There are 3 pink light bulbs, 6 blue light bulbs, and 3 red light bulbs. How many distinct orders of light bulbs are there if two light bulbs of the same color are considered identical (not distinct)?
Solution
To solve this problem, we can use the formula for permutations of a multiset. This formula is n! / (r1! * r2! * ... * rk!), where n is the total number of items, and r1, r2, ..., rk are the numbers of each type of item.
In this case, n = 12 (the total number of light bulbs), r1 = 3 (the number of pink light bulbs), r2 = 6 (the number of blue light bulbs), and r3 = 3 (the number of red light bulbs).
So, the number of distinct orders of light bulbs is 12! / (3! * 6! * 3!).
First, calculate the factorial of each number:
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 479,001,600 3! = 3 * 2 * 1 = 6 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Then, divide 12! by the product of 3!, 6!, and 3!:
479,001,600 / (6 * 720 * 6) = 220
So, there are 220 distinct orders of light bulbs.
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