QJEEAdv14Maa62c: JEE Advanced 2014 Mathematics - Class 12 JEEAdv14Maa62c Solution: Box contains three cards bearing numbers box contains five…

Given three Boxes :

Given, that a card is drow from each box and $x_i$ denote the number on the card drawn from $i^{\text {th }}$ box, $i=1,2,3$ .

Sample Space $={ }^3 \mathrm{C}_1 .{ }^5 \mathrm{C}_1 .{ }^7 \mathrm{C}_1=3 \cdot 5 \cdot 7=105$ .

Let B be the event when $x_1, x_2, x_3$ are in A.P.

$\Rightarrow 2 x_2=x_1+x_3$

Here, $x_1+x_3$ is even number and for every even value of $x_1+x_3$ there is only one possible value of $x_2$

$x_1+x_2$ is even in two cases

Case I: When both $x_1$ and $x_2$ are even

No. of ways $={ }^1 \mathrm{C}_1$ . ${ }^3 \mathrm{C}_{1=}=3$

Case II: When both $x_1$ and $x_2$ are odd

No. of ways $={ }^2 \mathrm{C}_1 .{ }^4 \mathrm{C}_1=2 \times 4=8$

Number of favorable cases for $\mathrm{B}=3+8=11$

Required probability $=\frac{11}{105}$

Hint:

$2 x_2=x_1+x_3$ indicate, $x_1+x_3$ is even and for every even value of $x_1+x_3$ there is only one possible value of $x_2$ . Now, $x_1+x_3$ is even in two cases:

(i) When both $x_1$ and $x_2$ are even.

(ii) When both $x_1$ and $x_2$ are odd.