To solve this problem, we need to calculate the probability of selecting 3 Regular and 3 BBQ chips out of the 10 selected bags.

Step 1: Determine the total number of ways to choose 10 chips from the box

The total number of chips in the box is:

$11 \text{ (Ketchup)} + 9 \text{ (BBQ)} + 11 \text{ (Sour Cream)} + 13 \text{ (Regular)} = 44 \text{ chips}$

The number of ways to choose 10 chips from 44 without regard to order is given by the combination formula:

$\text{Total ways} = \binom{44}{10}$

Step 2: Calculate the number of favorable outcomes

We need exactly 3 Regular and 3 BBQ chips. This means that out of the remaining 4 chips, some will be Ketchup and some will be Sour Cream.

• Number of ways to choose 3 Regular chips out of 13:

$\binom{13}{3}$

• Number of ways to choose 3 BBQ chips out of 9:

$\binom{9}{3}$

• Number of ways to choose the remaining 4 chips from the 22 Ketchup and Sour Cream chips (11 Ketchup + 11 Sour Cream):

$\binom{22}{4}$

The number of favorable outcomes is:

$\text{Favorable outcomes} = \binom{13}{3} \times \binom{9}{3} \times \binom{22}{4}$

Step 3: Calculate the probability

The probability is the ratio of the number of favorable outcomes to the total number of outcomes:

$P(\text{3 Regular and 3 BBQ}) = \frac{\binom{13}{3} \times \binom{9}{3} \times \binom{22}{4}}{\binom{44}{10}}$

Step 4: Calculate the combinations

Let's calculate each combination:

$\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286$ $\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$ $\binom{22}{4} = \frac{22 \times 21 \times 20 \times 19}{4 \times 3 \times 2 \times 1} = 7315$ $\binom{44}{10} = \frac{44 \times 43 \times \ldots \times 35}{10 \times 9 \times \ldots \times 1} = 7,379,783,045$

Step 5: Final Probability Calculation

$P(\text{3 Regular and 3 BBQ}) = \frac{286 \times 84 \times 7315}{7,379,783,045}$

Let's calculate this value.

$P(\text{3 Regular and 3 BBQ}) \approx \frac{176041560}{7,379,783,045} \approx 0.0238$

So, the probability that 3 of the 10 chips are Regular and 3 are BBQ is approximately $0.0238$ or 2.38%.

Would you like any details or have any questions?

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1. What is the probability of picking exactly 4 Ketchup chips?

2. How would the probability change if 5 chips were picked instead of 10?

3. What is the probability that all 10 chips are from one flavor?

4. How does the probability change if the number of BBQ chips is increased to 15?

5. What if we want to find the probability of selecting 2 Ketchup and 4 Sour Cream?

6. What is the probability that none of the selected chips are Regular?

7. How many possible selections include exactly 2 BBQ chips?

8. What is the probability of selecting at least one of each flavor?

Tip: When calculating probabilities involving combinations, remember to break the problem down into steps and check your work at each stage to avoid mistakes.