Question

1. toothpaste display how many different ways can 5 identical tubes of tartar control toothpaste, 3 identical tubes of bright white toothpaste, and 4 identical tubes of mint toothpaste be arranged in a grocery store counter display?

Explanation:

Step1: Calculate the total number of tubes

The total number of tubes is \(n=5 + 3+4=12\).

Step2: 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 \(\frac{n!}{n_1!n_2!\cdots n_k!}\). Here \(n = 12\), \(n_1=5\) (tubes of tartar control toothpaste), \(n_2 = 3\) (tubes of bright white toothpaste), \(n_3=4\) (tubes of mint toothpaste).
So the number of arrangements is \(\frac{12!}{5!3!4!}\).

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Answer:

\(27720\)