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.