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

Step 1: Fixed part first: type, bread and size give \(5 \times 4 \times 2 = 40\) base choices.
Step 2: Count sauce selections by first arranging in order, then dividing out the overcount, since order doesn't actually matter. One sauce: \(6\) ways directly. Two sauces: pick them in order (\(6 \times 5 = 30\) ways), then divide by \(2!\) since each pair is counted twice: \(30/2 = 15\). No sauce: \(1\) way.
Step 3: Total sauce options \(= 1 + 6 + 15 = 22\), so the total orders \(= 40 \times 22 = 880\).
\[ \boxed{880} \]