Knowing that you must choose some candy, determine the odds that at bulk barn, out of the 13 different types of possibile candies, you choose exactly 7.

To determine the odds that you choose exactly 7 types of candy out of 13 different types at Bulk Barn, we need to calculate the number of ways to choose 7 candies out of 13 and then compare it to the number of ways to choose any number of candies out of 13.

### Step-by-Step Solution:

1. **Calculate the Number of Ways to Choose Exactly 7 Candies:**

   The number of ways to choose 7 candies out of 13 is given by the combination formula \( \binom{13}{7} \):
   \[
   \binom{13}{7} = \frac{13!}{7!(13-7)!} = \frac{13!}{7! \cdot 6!}
   \]

   Calculate this step-by-step:
   \[
   \binom{13}{7} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 1716
   \]

2. **Calculate the Total Number of Possible Choices:**

   The total number of ways to choose any number of candies (from 1 to 13, since you must choose some candy) is the sum of the combinations from \( \binom{13}{1} \) to \( \binom{13}{13} \):
   \[
   \sum_{k=1}^{13} \binom{13}{k}
   \]

   This can be simplified using the binomial theorem, which states that the sum of all combinations from \( \binom{n}{0} \) to \( \binom{n}{n} \) is \( 2^n \). However, we need to exclude the case where no candy is chosen:
   \[
   2^{13} - \binom{13}{0} = 8192 - 1 = 8191
   \]

3. **Calculate the Probability:**

   The probability \( P \) of choosing exactly 7 candies out of 13 is:
   \[
   P = \frac{\binom{13}{7}}{\sum_{k=1}^{13} \binom{13}{k}} = \frac{1716}{8191}
   \]

4. **Convert Probability to Odds:**

   The odds are given by the ratio of the probability of the event occurring to the probability of the event not occurring:
   \[
   \text{Odds} = \frac{P}{1-P}
   \]

   Calculate \( 1 - P \):
   \[
   1 - P = 1 - \frac{1716}{8191} = \frac{8191 - 1716}{8191} = \frac{6475}{8191}
   \]

   Therefore, the odds are:
   \[
   \text{Odds} = \frac{\frac{1716}{8191}}{\frac{6475}{8191}} = \frac{1716}{6475} = \frac{858}{3237} \approx 1:3.775
   \]

So, the odds that you choose exactly 7 types of candy out of 13 are approximately \(1:3.775\).

Would you like more details or have any questions?

Here are some related questions:

1. What is the probability of choosing exactly 5 types of candy out of 13?
2. How many ways can you choose at least 10 types of candy out of 13?
3. What is the probability of choosing at most 4 types of candy out of 13?
4. If you choose exactly 8 types of candy, what is the probability that a specific type is included?
5. How many ways can you choose 7 types of candy if two specific types must be included?
6. What is the number of ways to choose an even number of types of candy out of 13?
7. How many ways can you choose exactly 3 types of candy out of 13 if one specific type must not be included?
8. What is the probability of choosing exactly 1 type of candy out of 13?

**Tip**: When working with combinations, remember that the order of selection does not matter. Use combinations (binomial coefficients) rather than permutations.

Explore the probability of choosing exactly 7 types of candy out of 13 at Bulk Barn. Learn how to calculate binomial coefficients and apply the binomial theorem to determine odds. Suitable for high school students and anyone interested in combinatorics and probability.

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