Question

In a cafeteria, there are 5 breads. One can choose 1 bread from the available breads, either small or large sized, and can choose up to 2 sauces from 6 available sauces. What is the number of different ways one can place an order?

Hint:

When a problem states "at most k" or "up to k" items can be chosen, you must sum the combinations for choosing 0, 1, 2, ..., up to k items.

Answer 1

Correct Answer: 220

Step 1: Understanding the Question:
This is a counting problem where an order consists of two independent choices: a bread and a selection of sauces. We need to find the total number of combinations by calculating the number of options for each choice and then multiplying them together.
Step 2: Key Formula or Approach:
- The Multiplication Principle: If a task consists of k steps, and the first step can be done in \(n_1\) ways, the second in \(n_2\) ways, ..., and the k-th in \(n_k\) ways, then the total number of ways to perform the task is \(n_1 \times n_2 \times \cdots \times n_k\).
- Combinations: The number of ways to choose r items from a set of n items is given by \(^nC_r = \frac{n!}{r!(n-r)!}\).
Step 3: Detailed Explanation:
Part A: Calculate the number of bread choices
There are 5 types of bread. Each type comes in 2 sizes (small or large).
Total bread choices = 5 types \(\times\) 2 sizes = 10 ways.
Part B: Calculate the number of sauce choices
There are 6 available sauces, and one can choose "up to 2 sauces". This means one can choose 0, 1, or 2 sauces.
- Number of ways to choose 0 sauces from 6 = \(^6C_0 = 1\).
- Number of ways to choose 1 sauce from 6 = \(^6C_1 = 6\).
- Number of ways to choose 2 sauces from 6 = \(^6C_2 = \frac{6 \times 5}{2 \times 1} = 15\).
Total sauce choices = \(1 + 6 + 15 = 22\) ways.
Part C: Calculate the total number of different orders
Using the multiplication principle, we multiply the number of bread choices by the number of sauce choices.
Total ways = (Number of bread choices) \(\times\) (Number of sauce choices)
Total ways = \(10 \times 22 = 220\).
Step 4: Final Answer:
The number of different ways one can place an order is 220.