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?

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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.