A delicatessen makes a roast-beef-on-sourdough sandwich for which you can choose from eight condiments.

(a) How many different types of roast-beef-on-sourdough sandwiches can the delicatessen prepare?
different types

(b) What is the minimum number of condiments the delicatessen must have available if it wishes to offers at least 1,000 different types of roast-beef-on-sourdough sandwiches?

### Part (a): How many different types of sandwiches can be prepared?

Given that there are 8 condiments to choose from, for each condiment, we can either:

- Include the condiment in the sandwich, or
- Not include the condiment.

Since each condiment has 2 possibilities (either included or not), and there are 8 condiments, the total number of different types of sandwiches is given by the total number of possible combinations of these condiments.

Thus, the number of different types of sandwiches is:

\[
2^8 = 256
\]

So, the delicatessen can prepare **256 different types of sandwiches**.

---

### Part (b): Minimum number of condiments for at least 1,000 types of sandwiches

We need to find the minimum number of condiments \( n \) such that the number of different types of sandwiches is at least 1,000. The number of different types of sandwiches is \( 2^n \), where \( n \) is the number of condiments.

We need to solve:

\[
2^n \geq 1000
\]

Taking the logarithm base 2 of both sides:

\[
n \geq \log_2(1000)
\]

Using logarithms:

\[
\log_2(1000) = \frac{\log_{10}(1000)}{\log_{10}(2)} = \frac{3}{0.3010} \approx 9.97
\]

Since \( n \) must be an integer, the minimum number of condiments is \( \lceil 9.97 \rceil = 10 \).

Thus, the delicatessen must have at least **10 condiments** to offer 1,000 or more different types of sandwiches.

---

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What if the delicatessen offers 12 condiments? How many sandwich combinations are possible?
2. How would the answer change if some condiments could be used in multiple amounts?
3. How many sandwiches can be made if you are limited to using at most 5 condiments out of the 8 available?
4. If the sandwich can have 2 types of bread, how many combinations are possible now?
5. What if you must include at least one condiment? How many valid sandwiches can you make?

**Tip:** Using logarithms is a handy way to determine powers of numbers when working with large quantities like combinations.

A delicatessen makes a roast-beef-on-sourdough sandwich for which you can choose from eight condiments.

(a) How many different types of roast-beef-on-sourdough sandwiches can the delicatessen prepare?
(b) What is the minimum number of condiments the delicatessen must have available if it wishes to offer at least 1,000 different types of roast-beef-on-sourdough sandwiches?

Explore how to calculate the number of roast-beef-on-sourdough sandwich combinations with 8 condiments using combinatorics, and find out the minimum number of condiments needed to offer at least 1,000 different sandwich types using logarithms. This problem is ideal for high school students learning exponents and logarithms.

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