Question

A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is:

• A. \(600\)
• B. \(840\)
• C. \(880\)
• D. \(800\)

Hint:

When a question says “up to $k$ items” from $n$ options, sum the combinations: \[ \sum_{r=0}^{k} \binom{n}{r}. \] Then multiply by the number of ways for all other independent choices.

Answer 1

The Correct Option is C

We count the number of choices at each step and multiply, since all choices are independent. Step 1: Choose sandwich type. There are 5 types: \[ 5 \text{ ways}. \]
Step 2: Choose bread. There are 4 breads: \[ 4 \text{ ways}. \]
Step 3: Choose size. Small or large: \[ 2 \text{ ways}. \]
Step 4: Choose sauces (up to 2 out of 6). The customer may take 0, 1, or 2 sauces (order doesn’t matter): \[ \binom{6}{0} = 1,\quad \binom{6}{1} = 6,\quad \binom{6}{2} = 15. \] Total ways to choose sauces: \[ 1 + 6 + 15 = 22. \]
Step 5: Multiply all choices. \[ \text{Total ways} = 5 \times 4 \times 2 \times 22 = 20 \times 44 = 880. \] Therefore, the number of different possible sandwich orders is \[ \boxed{880}. \]