Question

If two different numbers are taken from the set {0, 1, 2, 3,.... 10}, then the probability that their sum as well as absolute difference are both multiple of 4, is ?

• A. $\frac{6}{55}$
• B. $\frac{12}{55}$
• C. $\frac{14}{45}$
• D. $\frac{7}{55}$

Answer 1

The correct option is A $\frac{6}{55}$

Total number of ways os selecting 2 different numbers from $\{0,1,2,....10\}{=}^{11}{C}_2=55$
Let two numbers selected be x and y.
Then, x + y = 4m ...(i)
and x - y = 4n .....(ii)
$\Rightarrow 2x=4(m+n)and2y=4(m-n)$
$\Rightarrow x=2(m+n)andy=2(m-n)$
$\therefore$ x and y both are even numbers.
$\begin{array}{cc}x& y\\ 0& 4,8\\ 2& 6,10\\ 4& 0,8\\ 6& 2,10\\ 8& 0,4\\ 10& 2,6\end{array}$
But here arrangement of x & y does not matter. Only selection of two numbers is necessary. Therefore, total 6 combinations are possible
$\therefore$ Required probability $=\frac{6}{55}$