Question

The number of rational numbers $\frac{p}{q}$, where $p,q$ $\in$ $1,2,3,4,5,6$ is

• A. 23
• B. 32
• C. 36
• D. 63

Answer 1

The correct option is A 23

The given 6 digits can occupy the numerator and denominator places in ${}^6{P}_2=30$ ways. But out of them these sets $(\frac{1}{2},\frac{2}{4},\frac{3}{6})$, $(\frac{2}{1}$, $\frac{4}{2},\frac{6}{3})$, $(\frac{2}{3}$, $\frac{4}{6})$,$(\frac{3}{2},\frac{6}{4})$, $(\frac{1}{3},\frac{2}{6})$ and $(\frac{3}{1}$ , $\frac{6}{2})$ represent same number. In all above cases we have to consider only one case each.
$\therefore$ no. of rational numbers
$=30-(2+2+1+1+1+1)+1$
$=22+1$ ($1$ is a rational no)
$=23$