Question

A shop sells two kinds of rolls: egg roll and mutton roll. Onion, tomato, carrot, chilli sauce and tomato sauce are the additional ingredients. A customer can have any combination of additional ingredients, or a standard roll with no additional ingredients, subject to the following constraints:

(a) Tomato sauce can be added if the roll has egg, but not if it is a mutton roll.
(b) Chilli sauce can be added only if the roll has onion or tomato or both; otherwise it cannot be added.

How many different rolls can be ordered according to these rules?

• A. 21
• B. 33
• C. 40
• D. 42

Hint:

Split the count by roll type since tomato sauce depends on it, and handle chilli sauce separately since it depends only on onion and tomato being present.

Answer 1

The Correct Option is D

Step 1: Fix the roll type first.
Every order starts with a choice of egg roll or mutton roll, 2 base choices. Everything else is a set of optional add-ons chosen from onion, tomato, carrot, chilli sauce and tomato sauce, subject to the two rules.

Step 2: Count valid onion, tomato and chilli-sauce combinations first, since rule (b) links them.
Onion and tomato are each independently present or absent, giving 4 combinations: neither, onion only, tomato only, both.
Chilli sauce is allowed only when onion or tomato, or both, is present. So:
- Neither onion nor tomato: chilli sauce cannot be added, only 1 way.
- Onion only, tomato only, or both: chilli sauce is free to add or not, 2 ways each, giving \(3 \times 2 = 6\) ways.
Total onion/tomato/chilli-sauce combinations \(= 1 + 6 = 7\).

Step 3: Count carrot separately, since it has no restriction.
Carrot can be present or absent freely, giving 2 ways, independent of everything else.

Step 4: Handle tomato sauce using rule (a), which depends on the roll type.
For an egg roll, tomato sauce may be added or not, 2 free ways. For a mutton roll, tomato sauce is never allowed, so only 1 way, absent.

Step 5: Multiply the independent counts for each roll type.
Egg roll: \(7 \times 2 \times 2 = 28\).
Mutton roll: \(7 \times 2 \times 1 = 14\).

Step 6: Add the two roll types together.
Total number of different rolls \(= 28 + 14 = 42\).

Final Answer:
The count of 42 matches option D. The smaller values 21, 33 and 40 come from forgetting to double for both roll types or mishandling the chilli sauce restriction. \[ \boxed{42} \]