Question

Two different numbers are taken from the set $\{0,1,2,3,4,5,6,7,8,9,10\}$. The probability that their sum and absolute value of difference both are multiple of $4$ is $\frac{x}{55}$, then $x$ equals

Answer 1

Total number of ways,
$n\left(S\right)={}^{11}{\text{C}}_2=55$
Let set $A=\{0,4,8\}$ and set $B=\{2,6,10\}$
If both the numbers are taken from either $A$ or $B$, then we have the favourable cases
Number of favourable cases = $\left({}^3{\text{C}}_2{+}^3{\text{C}}_2\right)=6$
$\therefore$ Required probability
$=\frac{6}{55}=\frac{x}{55}\Rightarrow x=6$