A company that makes crayons is trying to decide which 4 colors to include in a promotional mini-box of 4 crayons. The company can choose the 4 mini-box colors from its collection of 65 colors. How many mini-boxes are possible?

This problem involves finding how many ways we can choose 4 colors from a collection of 65 colors. Since the order in which we choose the colors doesn't matter, we need to use the **combination formula**.

The formula for combinations is given by:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

where:
- \( n \) is the total number of items (in this case, 65 colors),
- \( r \) is the number of items to choose (in this case, 4 colors),
- \( ! \) denotes factorial (the product of all positive integers up to that number).

Substituting the values into the formula:

\[
C(65, 4) = \frac{65!}{4!(65 - 4)!} = \frac{65!}{4! \times 61!}
\]

We can cancel the \( 61! \) from the numerator and denominator, which simplifies to:

\[
C(65, 4) = \frac{65 \times 64 \times 63 \times 62}{4 \times 3 \times 2 \times 1}
\]

Now let's calculate this:

\[
65 \times 64 = 4160
\]
\[
4160 \times 63 = 262080
\]
\[
262080 \times 62 = 16248960
\]
\[
4 \times 3 = 12
\]
\[
12 \times 2 = 24
\]
\[
\frac{16248960}{24} = 677040
\]

Therefore, the number of possible mini-boxes is:

\[
677,040
\]

### Answer:
The company can create **677,040** different mini-boxes of 4 crayons.

Would you like further details on how this was calculated or have any questions?

---

Here are some related questions you might find interesting:
1. What if the company wanted to create a mini-box with 5 crayons instead of 4?
2. How many combinations would there be if the company had only 50 colors available instead of 65?
3. How many ways could they select 3 colors from the 65?
4. What if the order of colors mattered? How would the number of possible mini-boxes change?
5. How would the solution change if we had to pick a specific color for the first crayon?

**Tip**: In problems where order doesn't matter, use combinations. When order matters, use permutations.

Discover how to calculate the number of possible combinations when choosing 4 crayons from a set of 65 colors. This combinatorics problem involves using the combination formula and covers key mathematical concepts. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation