CAT 2025

Slot 1

QA

Question & Solution

1. Concept Used

• Topic: Permutations and Combinations — Multiplication Principle & Combinatorial Counting
• Formula: $$ \text{Total Ways} = \binom{n_1}{1} \times \binom{n_2}{1} \times \binom{n_3}{1} \times \left( \binom{n_4}{0} + \binom{n_4}{1} + \binom{n_4}{2} \right) $$

2. Calculation

We break the order into independent choices and apply the Multiplication Principle at the end.

Step 1 — Choose a sandwich type: There are 5 types, and exactly one must be chosen. This gives $$\binom{5}{1} = 5$$ ways.

Step 2 — Choose a bread: There are 4 bread options, and exactly one must be chosen. This gives $$\binom{4}{1} = 4$$ ways.

Step 3 — Choose the size: The sandwich is either small or large — exactly one must be chosen. This gives $$\binom{2}{1} = 2$$ ways.

Step 4 — Choose sauces (optional, up to 2 from 6): The word "optionally" and "up to 2" means the customer may add 0, 1, or 2 sauces. So we sum all valid cases: $$\binom{6}{0} + \binom{6}{1} + \binom{6}{2} = 1 + 6 + 15 = 22 \text{ ways}$$

Step 5 — Multiply all independent choices together: $$\text{Total} = 5 \times 4 \times 2 \times 22 = 5 \times 4 \times 44 = 5 \times 176 = 880$$

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3. Solution

Answer = Option A (880) ✅

The total number of different ways in which an order can be placed for a sandwich is 880.