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Chapter 7: Problem 383

A manufacturer produces 7 different items. He packages assortments of equal parts of 3 different items. How many different assortments can be packaged?

Short Answer

Expert verified
The manufacturer can create 35 different assortments using equal parts of 3 different items from the available 7 items. This was determined using the combinations formula: \(C(7, 3) = \frac{7!}{3!(7-3)!} = 35\).

Step by step solution

01

Understand the Combinations Formula

The combinations formula is used to determine the number of ways to choose k items from a set of n items without considering the order. The formula for combinations is: \[C(n, k) = \frac{n!}{k!(n-k)!}\] where: - C(n, k) is the number of combinations - n is the total number of items in the set - k is the number of items to be chosen from the set - ! denotes the factorial of a number
02

Apply the Combinations Formula to the Problem

In this exercise, we have a set of 7 items (n = 7) and want to choose 3 of them (k = 3) to create an assortment. Therefore, we can use the combinations formula to find the number of different assortments: \[C(7, 3) = \frac{7!}{3!(7-3)!}\]
03

Calculate the Factorial Values

Next, we need to calculate the factorial values for 7, 3, and (7-3): 1. \(7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 5040\) 2. \(3! = 3 脳 2 脳 1 = 6\) 3. \((7-3)! = 4! = 4 脳 3 脳 2 脳 1 = 24\)
04

Substitute the Factorial Values into the Combinations Formula

Now, we can substitute the factorial values we calculated in Step 3 into the combinations formula: \[C(7, 3) = \frac{5040}{6 脳 24}\]
05

Calculate the Number of Combinations

Finally, we can calculate the number of different assortments: \[C(7, 3) = \frac{5040}{144} = 35\] There are 35 different assortments that can be packaged by the manufacturer using equal parts of 3 different items from the 7 available items.

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