A catering service offers 8 ​appetizers, 11 main​ courses, and 9 desserts. A banquet committee is to select 6 ​appetizers, 7 main​ courses, and 2 desserts. How many ways can this be​ done?


To determine how many ways the banquet committee can select 6 appetizers, 7 main courses, and 2 desserts, we use the concept of combinations (since the order of selection doesn't matter). The number of ways to choose \( r \) items from \( n \) items is given by the combination formula:

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

Where:
- \( n \) is the total number of items,
- \( r \) is the number of items to be selected.

### Step 1: Calculate combinations for appetizers
There are 8 appetizers, and the committee needs to select 6. So, the number of ways to choose 6 appetizers from 8 is:

\[
\binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8!}{6!2!} = \frac{8 \times 7}{2 \times 1} = 28
\]

### Step 2: Calculate combinations for main courses
There are 11 main courses, and the committee needs to select 7. The number of ways to choose 7 main courses from 11 is:

\[
\binom{11}{7} = \frac{11!}{7!(11-7)!} = \frac{11!}{7!4!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330
\]

### Step 3: Calculate combinations for desserts
There are 9 desserts, and the committee needs to select 2. The number of ways to choose 2 desserts from 9 is:

\[
\binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36
\]

### Step 4: Total number of ways
The total number of ways to select the appetizers, main courses, and desserts is the product of the individual combinations:

\[
\text{Total ways} = \binom{8}{6} \times \binom{11}{7} \times \binom{9}{2} = 28 \times 330 \times 36
\]

Now, let's calculate the result:

\[
\text{Total ways} = 28 \times 330 \times 36 = 332,640
\]

### Conclusion:
The total number of ways the banquet committee can make the selection is **332,640**.

Would you like more details on any step?

Here are 5 related questions:
1. How do combinations differ from permutations?
2. What is the formula for calculating combinations in general?
3. Can we extend this method to more complex selection scenarios?
4. How would the result change if the order of selection mattered?
5. How do factorials simplify large combination calculations?

**Tip:** Remember, combinations count selections where order doesn't matter, unlike permutations!

A catering service offers 8 appetizers, 11 main courses, and 9 desserts. A banquet committee is to select 6 appetizers, 7 main courses, and 2 desserts. How many ways can this be done?

Explore how to calculate the number of ways a banquet committee can select 6 appetizers, 7 main courses, and 2 desserts from a catering service offering 8 appetizers, 11 main courses, and 9 desserts. Learn about combinations, factorials, and step-by-step calculations. Suitable for high school students.

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