A candy manufacturer produces Halloween surprise bags by filling bags with 5 different surprises. How many different surprise bags can the company create if it stocks 14 different types of

To determine how many different Halloween surprise bags can be made, we are choosing 5 surprises from 14 different types of surprises. Since the order in which the surprises are chosen doesn't matter, and each type of surprise can be chosen only once, this is a combination problem.

The formula for combinations is:

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

where \( n \) is the total number of types of surprises (14 in this case) and \( r \) is the number of surprises to choose (5 in this case).

Substituting into the formula:

\[
\binom{14}{5} = \frac{14!}{5!(14-5)!} = \frac{14!}{5!9!}
\]

Now, let's calculate this.

### Simplification:

\[
\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1}
\]

Let me compute that value.The number of different Halloween surprise bags the company can create is \( 2002 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this problem:
1. How would the number of surprise bags change if 6 surprises were chosen instead of 5?
2. What happens if the company stocks 20 different types of surprises instead of 14?
3. How can we calculate the number of surprise bags if some surprises can be repeated in each bag?
4. What if we need to create surprise bags with at least one type of every surprise in stock?
5. How would the result change if the order of the surprises mattered?

**Tip:** Combination problems deal with selections where the order doesn't matter, while permutations handle cases where the order of selection does matter.

A candy manufacturer produces Halloween surprise bags by filling bags with 5 different surprises. How many different surprise bags can the company create if it stocks 14 different types of surprises?

Learn how to calculate the number of different Halloween surprise bags that can be created by choosing 5 surprises from 14 available types. This problem introduces the concept of combinations using the combination formula C(n, r). Suitable for high school students in Grades 9-12.

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