Question

If $C$ is the set of letters in the word 'cooler' find number of its proper subsets.

Answer 1

Set $C=\{c,o,l,e,r\}$
Elements in the set are unique, so we do not repeat the letters.

Thus, $o$ is written only once.
Now, $n\left(C\right)=5$ as there are $5$ elements in the set.

We can form $2^n$ subsets of a set containing $n$ elements.
Hence, $2^5=32$ subsets can be formed for set $C$.
However, one of these subsets will be $\{c,o,l,e,r\}$ which has all elements of set $C$.

Such subset is not a proper subset.
Hence, $2^5-1=32-1=31$ proper subsets can be formed for set $C$.