JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Permutations and Combinations

Let

S = {1, 2, 3, \dots, 100}

. The number of non-empty subsets A of S such that the product of elements in A is even is:

(A) $2^{100} - 1$
(B) $2^{50} (2^{50} - 1)$ ✓ Correct
(C) $2^{50} - 1$
(D) $2^{50} + 1$

Answer: (B)

Solution

A product is even precisely when the subset contains at least one even number, so it is easiest to count the opposite, subsets made only of odd numbers, and subtract. The set has

50

odd and

50

even numbers. The non-empty subsets number

2^{100} - 1

, and the subsets made purely of odd numbers number

2^{50}

, counting the empty set. Removing the all-odd subsets leaves

2^{100} - 2^{50}

, which factors as

2^{50} (2^{50} - 1)

JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Single correct option

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