**Choose the appropriate equation for the following set of data:**
 
Richmond Cafe allows guests to create their own pasta dishes.  For each dish, the guest picks one pasta from a choice of five, one sauce from a choice of six, up to three vegetables from a choice of nine, and one meat from a choice of five.  How many pasta dishes can a guest create if two vegetables are chosen and one item from each of the other categories

**Choose the appropriate equation for the following set of data:**
 
Richmond Cafe allows guests to create their own pasta dishes.  For each dish, the guest picks one pasta from a choice of five, one sauce from a choice of six, up to three vegetables from a choice of nine, and one meat from a choice of five.  How many pasta dishes can a guest create if two vegetables are chosen and one item from each of the other categories

𝑁=𝐶(5,1)⋅𝐶(6,1)⋅𝐶(9,2)⋅𝐶(5,1)

𝑁=𝑃(5,1)⋅𝑃(6,1)⋅𝑃(9,2)⋅𝑃(5,1)

𝑁=𝐶(5,1)⋅𝐶(6,1)⋅𝐶(9,3)⋅𝐶(5,1)

𝑁=𝐶(25,5)

𝑁=25

To determine the appropriate equation for calculating the number of possible pasta dishes that can be created given the options and constraints, let's break down the problem step by step:

1. **Pasta Choice**: 
   - The guest picks one pasta from a choice of five.
   - The number of ways to do this is \( \text{C}(5, 1) \), where \( \text{C}(5, 1) \) represents the number of combinations of 5 items taken 1 at a time.

2. **Sauce Choice**: 
   - The guest picks one sauce from a choice of six.
   - The number of ways to do this is \( \text{C}(6, 1) \), representing the number of combinations of 6 items taken 1 at a time.

3. **Vegetable Choices**: 
   - The guest picks 2 vegetables from a choice of nine.
   - The number of ways to do this is \( \text{C}(9, 2) \), representing the number of combinations of 9 items taken 2 at a time.

4. **Meat Choice**: 
   - The guest picks one meat from a choice of five.
   - The number of ways to do this is \( \text{C}(5, 1) \), representing the number of combinations of 5 items taken 1 at a time.

The total number of possible pasta dishes is the product of all these individual choices:

\[
N = \text{C}(5, 1) \cdot \text{C}(6, 1) \cdot \text{C}(9, 2) \cdot \text{C}(5, 1)
\]

### Conclusion:
The correct equation from the options given is:

\[
\boxed{N = \text{C}(5, 1) \cdot \text{C}(6, 1) \cdot \text{C}(9, 2) \cdot \text{C}(5, 1)}
\]

Would you like to go through any specific part of this process, or do you have other questions?

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**Relative Questions:**
1. How do you calculate combinations \( \text{C}(n, k) \) for any values of \( n \) and \( k \)?
2. What is the difference between permutations and combinations?
3. How would the formula change if the guest could choose up to three vegetables instead of exactly two?
4. What if the guest could choose multiple items from the meat category? How would the equation change?
5. Can you calculate the exact number of possible pasta dishes using the selected equation?
6. How would the calculation differ if the guest could choose a different number of sauces?
7. Why is it important to distinguish between combinations and permutations in this context?
8. How does the concept of "no repetition allowed" apply to this problem?

**Tip:** Remember that combinations are used when the order of selection does not matter, while permutations are used when the order does matter.

Explore how to calculate the number of possible pasta dishes at Richmond Cafe using combinations. Learn about the mathematical concept of combinations and apply it to different categories like pasta, sauce, vegetables, and meat choices. Suitable for Grades 7-9.

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