KVPY SX SB Exam 19-Nov-2017 Question Paper

We have,
m, n be two distinct integer chosen
randomly from the set
∴ Total number of outcomes
$4^{\mathit{m}}+4^{\mathit{n}}+3$
${\left(5-1\right)}^{\mathit{m}}+{\left(5-1\right)}^{\mathit{n}}+3$
$5\mathit{K}+{\left(-1\right)}^{\mathit{m}}+5\mathit{\lambda }+{\left(-1\right)}^{\mathit{n}}+3$
$5\left(\mathit{K}+\mathit{\lambda }\right)+{\left(-1\right)}^{\mathit{m}}+{\left(-1\right)}^{\mathit{n}}+3$
$4^{\mathit{m}}+4^{\mathit{n}}+3$ is divisible by 5 if $\mathit{m}$ and $\mathit{n}$ both are even.

∴ Favourable outcomes $=50{\mathit{C}}_2$
Required probability $=\frac{50{\mathit{C}}_2}{100{\mathit{C}}_2}$
$=\frac{50\times 49}{100\times 99}=\frac{49}{198}=0.247$
$0.247\in \left(0,0.25\right]$