A shop sells two kinds of rolls- egg roll and mutton roll. Onion, tomato, carrot, chili sauce and tomato sauce are the additional ingredients. You can have any combination of additional ingredients, or have standard rolls without any additional ingredients subject to the following constraints:

(a) You can have tomato sauce if you have an egg roll, but not if you have a mutton roll.

(b) If you have onion or tomato or both you can have chilli sauce, but not otherwise.

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

The answer is 42.

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Let 5 additional ingredients Onion, Tomato, Carrot, Chili Sauce and Tomato Sauce are denoted by O, T, C, CS, TS respectively.

Case I: Egg Roll: Without any restriction the number of ways ‘Egg Roll’ can be Ordered: =2×2×2×2×2=32 ways (As each of the five additional ingredients can be selected or rejected i.e. 2 ways) The cases are: (standard ‘Egg Roll’), (O), (T), (C), (CS), (TS), (O, T), (O, C) … (O, T, C, CS, TS)

Out of these cases the following four cases are not possible by the condition (b) as given in the question: (CS) (CS, TS) (CS, C) (CS, C, TS) Total number of ways Egg Roll can be ordered =32–4=28

Case II: Mutton Roll: Total number of cases for Mutton roll must be half of the total numbers of cases for Egg Roll as mutton roll will never have the ingredient TS. Total cases for Mutton Roll =28/2=14 Required number of cases =28+14= 42