Question

a sales assistant is hanging 13 sweaters on a rack. he has 3 green sweaters, 4 gray sweaters, and 6 white sweaters. in how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)?

Explanation:

Step1: Use the formula for permutations of multi - sets

The formula for permutations of a multi - set with \(n\) objects, where there are \(n_1\) of one kind, \(n_2\) of another kind, \(\cdots\), \(n_k\) of the \(k\)th kind is \(\frac{n!}{n_1!n_2!\cdots n_k!}\), where \(n=n_1 + n_2+\cdots + n_k\).
Here, \(n = 13\) (total number of sweaters), \(n_1=3\) (green sweaters), \(n_2 = 4\) (gray sweaters), \(n_3=6\) (white sweaters).

Step2: Calculate the factorial values

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So, \(13! = 13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\), \(3! = 3\times2\times1=6\), \(4! = 4\times3\times2\times1 = 24\), \(6! = 6\times5\times4\times3\times2\times1=720\).

Step3: Substitute into the formula

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Answer:

\(60060\)