Question

Assertion :Out of 5 marbles consecutively numbered, three are drawn randomly, the probability that numbers on them in A.P. is $\frac{2}{5}$ Reason: Out of $2x+1$ marbles consecutively numbered three are drawn randomly, the probability that the number on them are in A.P. is $\frac{3n}{4n^2-1}$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

For Number of marbles $2n+1=5$
$\therefore n=2$
$\therefore P\left(A\right)=\frac{3n}{4n^2-1}=\frac{3\times 2}{15}=\frac{2}{5}$
For $a,b,c$ in $A.P.,2b=a+c$ which means $a,c$ both are either even or odd
$\therefore a$ and $c$ can be chosen in ${}^n{\text{C}}_2{+}^{n+1}{\text{C}}_2$ $=\frac{n(n-1)}{2}+\frac{(n+1)n}{2}$
$P\left(A\right)=\frac{n^2}{{}^{2n+1}{\text{C}}_3}=\frac{n^2\times 6}{(2n+1)\left(2n\right)(2n-1)}$ $=\frac{3n}{4n^2-1}$