Mathematics - Permutation & Combination Question with Solution | TestHub

Let $\mathrm{m}=6\mathrm{a},\mathrm{n}=6\mathrm{b}$
$\mathrm{m}<\mathrm{n}\Rightarrow \mathrm{a}<\mathrm{b}$
where $\mathrm{a}\&\mathrm{b}$ are co-prime numbers also since $m\&n$ are 2 digit nos, so
$10\le \mathrm{m}\le 99\&10\le \mathrm{n}\le 99$
i.e. $2\le \mathrm{a}\le 16\&2\le \mathrm{b}\le 16$
( $\because$ a is integer)
Now
$2\le \mathrm{a}<\mathrm{b}\le 16\&\mathrm{a}\&\mathrm{b}$ are co-prime
$\therefore$ if
$\mathrm{a}=2,\mathrm{b}=3,5,7,9,11,13,15$
$a=3,b=4,5,7,8,10,11,13,14,16$
$a=4,b=5,7,9,11,13,15$
$a=5,b=6,7,8,9,11,12,13,14,16$
$\mathrm{a}=6,\mathrm{b}=7,11,13$
$\mathrm{a}=7,\mathrm{b}=8,9,10,11,12,13,15,16$
$\mathrm{a}=8,\mathrm{b}=9,11,13,15$
$a=9,b=10,11,13,14,16$
$\mathrm{a}=10,\mathrm{b}=11,13$
$\mathrm{a}=11,\mathrm{b}=12,13,14,15,16$
$\mathrm{a}=12,\mathrm{b}=13$
$a=13,b=14,15,16$
$\mathrm{a}=14,\mathrm{b}=15$
$\mathrm{a}=15,\mathrm{b}=16$
64 ordered pairs