Question

Let $p,q$ be chosen one by one from the set $\{1,\sqrt{2},\sqrt{3},2,e,\pi \}$ with replacement. Now a circle is drawn taking $(p,q)$ as its centre. Then the probability that at the most two rational points exist on the circle is (rational points are those points whose both the coordinates are rational)

• A. $2/3$
• B. $7/8$
• C. $8/9$
• D. none of these

Answer 1

The correct option is C $8/9$

Given set $\{1,\sqrt{2},\sqrt{3},2,e,\pi \}$

Probability of the no. choosen from the set with replacement $=4\times {}^5{C}_1=20$

$\Rightarrow p$ (at most $2$ rational no's ) $=\frac{4\times {}^5{C}_1}{{}^5{C}_1+{}^5{C}_2}$

$\Rightarrow \frac{20}{5+10}=\frac{4}{3}\times \frac{{}^5{C}_2}{{}^5{C}_3}$

$=\frac{8}{9}.$

Hence, the answer is $\frac{8}{9}.$