Question

Let $S$ be a set of all distinct numbers of the form $\frac{P}{Q}$ , where $p,q\in \left\{1,2,3,4,5,6\right\}$. What is the cardinality of the set $S$?

• A. $21$
• B. $23$
• C. $32$
• D. $36$

Answer 1

The correct option is B

$23$

Explanation For The Correct Option:

Finding is the cardinality of the set $S$

The number of elements taken at a time $=2$

So, the total possible numbers of form $\frac{p}{q}$​ when $p\ne q$is $=2{}^6{C}_2=30$

Possible numbers when $p=q$ is ${=}^6{C}_1=6$

Therefore the total numbers is $=30+6=36$

$\frac{1}{1}=\frac{2}{2}=\frac{3}{3}=\frac{4}{4}=\frac{5}{5}=\frac{6}{6}$

So subtracting $5$ from the cardinality of the set.

$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}$ [subtract $2$ from the cardinality of the set]

$\frac{2}{1}=\frac{4}{2}=\frac{6}{3}$ [subtracting $2$ from the cardinality of the set]

$\frac{1}{3}=\frac{2}{6}$ [subtracting $1$ from the cardinality of the set]

$\frac{3}{1}=\frac{6}{2}$ [subtracting $1$ from the cardinality of the set]

$\frac{2}{3}=\frac{4}{6}$ [subtracting $1$ from the cardinality of the set]

$\frac{3}{2}=\frac{6}{4}$ [subtracting $1$ from the cardinality of the set]

So, the cardinality of set $=36-5-2-2-1-1-1-1=23.$

Hence, the correct option is (B)