Question

Let $S$ be a set of all distinct numbers of the form $\frac{p}{q}$, where $p,q$ $\in [1,2,3,4,5,6]$. What is the caardinality of the set $S$?

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

Answer 1

The correct option is B $23$

Total possible numbers of form $\frac{p}{q}$ when $p\ne q$ is $={}^6{C}_2=30$

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

Therefore total numbers $30+6=36$

$\frac{1}{1}=\frac{2}{2}=\frac{3}{3}=\frac{4}{4}=\frac{5}{5}=\frac{6}{6}$ (five numbers deducted from caardinality of set)

$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}$ (two numbers deducted from caardinality of set)

$\frac{2}{1}=\frac{4}{2}=\frac{6}{3}$ (two more numbers deducted from caardinality of set)

$\frac{1}{3}=\frac{2}{6}$ (one number deducted from caardinality of set)

$\frac{3}{1}=\frac{6}{2}$ (one more number deducted from caardinality of set)

$\frac{2}{3}=\frac{4}{6}$ (one number deducted from caardinality of set)

$\frac{3}{2}=\frac{6}{4}$ (one more number deducted from caardinality of set)

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

So, option B is correct.