A sales assistant is hanging 12 sweaters on a rack. He has 5 purple sweaters, 5 pink sweaters, and 2 gray sweaters. In how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)?
Question
A sales assistant is hanging 12 sweaters on a rack. He has 5 purple sweaters, 5 pink sweaters, and 2 gray sweaters. In how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)?
Solution
This is a permutation problem involving identical objects. The formula for permutations of n objects where there are n1 of one type, n2 of another, etc. is:
n! / (n1! * n2! * ... * nk!)
In this case, n (the total number of sweaters) is 12. There are 5 purple sweaters (n1), 5 pink sweaters (n2), and 2 gray sweaters (n3).
So, we plug these numbers into the formula:
12! / (5! * 5! * 2!)
Calculating factorials, we get:
479001600 / (120 * 120 * 2)
Solving this, we find that there are 1,814,400 distinct ways to arrange the sweaters.
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