Question

A set S has $7$ elements. How many subsets having at most $5$ elements does it have?
$\left[5\right]$

Answer 1

Number of subsets of $S=n\left(P\right(S\left)\right)$
$n\left(S\right)=7$
$\Rightarrow n\left(P\right(S\left)\right)=27=128$
$\left[1\right]$

The number of $7$ element subsets of $S=1$ (S itself).
$\left[1\right]$
$6$ element subsets of $S$ are obtained by removing $1$ element in $S$ and this can be done in $7$ ways.
The number of $6$ element subsets of $S=7$.
$\left[1\right]$
The total number of subsets having at most $5$ elements = total number of subsets - the number of $6$ or $7$ element subsets.
The number of subsets of S with at most $5$ elements $=128-7-1=120$
$\left[2\right]$