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: Start from the unrestricted count and subtract the invalid cases.
Ignoring both rules for a moment, each of the 5 add-ons, onion, tomato, carrot, chilli sauce and tomato sauce, is independently present or absent, giving $2^5 = 32$ combinations for each roll type.

Step 2: Remove the mutton roll cases that break rule (a).
Rule (a) says tomato sauce is not allowed on a mutton roll. Among the 32 mutton-roll combinations, exactly half have tomato sauce present, since it is one independent yes or no bit: $32 / 2 = 16$ invalid mutton combinations. Valid mutton combinations so far: $32 - 16 = 16$.

Step 3: Remove the cases that break rule (b), separately for each roll type.
Rule (b) forbids chilli sauce when both onion and tomato are absent. For the egg roll, fixing onion = no, tomato = no, chilli sauce = yes leaves carrot and tomato sauce free, giving $2 \times 2 = 4$ invalid combinations. Valid egg combinations: $32 - 4 = 28$.
For the mutton roll, tomato sauce is already fixed at no from Step 2, so within those 16 combinations, the invalid rule (b) cases have onion = no, tomato = no, chilli sauce = yes, tomato sauce = no, with carrot free: $2$ invalid combinations. Valid mutton combinations: $16 - 2 = 14$.

Step 4: Add the two valid counts.
Total valid rolls $= 28 + 14 = 42$.

Final Answer:
This subtraction approach also gives 42, confirming option D. \[ \boxed{42} \]