First slide

Combinations

Question

The number of subsets of the set $A = \{a_{1}, a_{2}, \dots, a_{n}\}$ which contain even number of elements is

Moderate

A

$2^{n - 1}$
B

$2^{n} - 1$
C

$2^{n - 2}$
D

$2^{n}$

Solution

For each of the first $(n - 1)$ elements $a_{1}, a_{2}$ …. $\dots, a_{n - 1}$ we have two choices:
either $a_{i} (1 \leq i \leq n - 1)$ lies in the subset or $a_{i}$ doesn’t lie in the subset.
For the last element we have just one choice. If even number of elements have already been selected,
we do not include $a_{n}$ in the subset otherwise (when odd number of elements have been selected)
we include it in the subset Thus, the number of subsets of $A = \{a_{1}, a_{2}, \dots, a_{n}\}$
which contain even number of elements is equal to $2^{n - 1}$

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