Question

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

The manufacturer can create 35 different assortments by selecting 3 different items out of 7. This is determined using the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n = 7 \) (total items) and \( r = 3 \) (items being selected). Substituting the values and performing the calculations reveals that there are 35 possible combinations of 3 items from 7.

Step-by-step solution

Understand the concept of combination

A combination is a selection of items from a larger set, where the order of the items does not matter and every item is distinct. In this exercise, we are trying to find the number of ways to choose 3 items out of 7. We can represent this as the combination C(n, r), where n is the total number of items and r is the number of items being chosen. In our case, we have n = 7 and r = 3.

Apply the formula for combinations

The formula for calculating combinations is C(n, r) = \( \frac{n!}{r!(n-r)!} \), where n! (n factorial) represents the product of all positive integers up to n. Using the formula, we can calculate the number of different assortments the manufacturer can create.

Plug the values into the formula

We have n = 7 and r = 3, so the number of combinations is C(7, 3) = \( \frac{7!}{3!(7-3)!} \). Now, let's calculate the factorials: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24

Calculate the number of combinations

Now, plug the values of the factorials into the formula: C(7, 3) = \( \frac{5040}{6 \times 24} \). Divide the factorials: C(7, 3) = \( \frac{5040}{144} \) Finally, divide the numerator by the denominator: C(7, 3) = 35

Interpret the result

The manufacturer can create 35 different assortments by selecting 3 different items out of 7.

Key concepts

Combination Formula

The combination formula is an essential tool in mathematics, especially in statistics and probability, for determining how many different ways a specific number of items can be chosen from a larger pool. This formula is crucial when the order in which the items are selected doesn't matter. For instance, if you're creating a fruit salad and want to know how many different ways you can select 3 types of fruit from a selection of 7, you'd use the combination formula.

The formula is given by:\[\begin{equation}C(n, r) = \frac{n!}{r!(n-r)!}\end{equation}\]where:

• \( C(n, r) \) represents the number of combinations,
• \( n \) is the total number of items,
• \( r \) is the number of items to choose, and
• The exclamation point \( ! \) denotes a factorial.

To apply this in the given exercise, we calculate the number of ways to package 3 different items from a set of 7. This is written as \( C(7, 3) \), and using the combination formula, we compute this to be 35 different assortments.

Factorial

In the realm of mathematics, the factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It's a fundamental component in permutations, combinations, and various functions. The factorial function grows very rapidly with the value of \( n \).

For example, the factorial of 3, expressly written as \( 3! \), is computed by multiplying all positive integers up to 3:\[\begin{equation}3! = 3 \times 2 \times 1 = 6\end{equation}\]It's important to note that \( 0! \) is specially defined as 1, which often comes in handy during calculations involving combinations and permutations. As factorials increase with \( n \), the numbers become quite large, making a factorial calculator or computational software useful for larger values.

Permutations and Combinations

When exploring the different ways in which a set of objects can be arranged or selected, two fundamental concepts are permutations and combinations. Both deal with selecting items, but they differ in whether the order of selection is important.

Permutations refer to the arrangement of all or part of a set of objects, with order being taken into account. A popular example of a permutation problem would be arranging letters in a word. For instance, the word 'CAT' can be arranged in 6 different ways (CAT, CTA, ACT, ATC, TAC, TCA).
Combinations, on the other hand, are selections of items where order does not matter. They are often used in scenarios such as lottery drawings or when creating sets where positioning is irrelevant. A simple example is selecting a committee from a larger group, where who gets picked first doesn't carry any weight.
Understanding these two concepts can be fundamentally important for solving a wide variety of problems in probability theory, statistical modeling, and decision-making situations. They both utilize factorials in their mathematical formulations, and distinguishing between them is key to proper application in real-world scenarios.