A company interviewed 5 applicants each for the posts of the Director

I used a different way.

Let's imagine there are 5 different people. Then number of ways is 5*5=25

But Jack and jill cant be elected for both positions. Number of ways to do this is 2.

25-2 =23
Now we need to subtract 2 cases where

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My Understanding is as such :-
Applicants for D - 3 +2 (Jack and Jill)
Applicants for P - 3 +2 (Jack and Jill)

If Jack/Jill is not selected as D, then D has 3 choices, and P has 5 choices.
Therefore, no. of choices = 3*5

If Jack/Jill is selected as D, P has 4 choices.
Therefore no. of choices = 2*4

So, total number of ways for selection = 3 *5 + 2*4 = 15+8 = 23

Let me know if this seems correct

Simplification of the question can be .... There are 2 common candidates in different sets. So unique candidates in each group are 3

D P
3 * 3 = 9
Jack *3 = 3
Jill *3 = 3
3 * Jack = 3
3 * Jill = 3
Jack * Jill = 1
Jill * Jack =1

Total=23

Hope ...I'm correct!...

Jack = X, Jill = Y

CASES:
X for D and Y for P: 1*1=1
Y for D and X for P: 1*1=1
X for D and ≠Y for P: 1*3C1=3
X for P and ≠Y for D: 1*3C1=3
Y for D and ≠X for P: 1*3C1=3
Y for P and ≠X for D: 1*3C1=3
Not X or Y for any: 3C1*3C1=9

Total: 2+3(4)+9=2+12+9=23

Ans (B)

If both Jack and Jill are selected, then there are 2 ways: Jack is the Director and Jill is the President, and vice versa.

If Jack is selected and Jill is not, then there are 1 x 3 = 3 ways if Jack is selected as the Director and another 3 ways if he is selected as the President. Therefore, there are a total of 6 ways if Jack is selected and Jill is not.

Likewise, there are a total of 6 ways if Jill is selected (as either the Director or the President) and Jill is not (selected at all).

Finally, if neither Jack nor Jill is selected, then there are 3 x 3 = 9 ways for the Director and the President to be chosen.

Therefore, there are a total of 2 + 6 + 6 + 9 = 23 ways.

Alternate Solution:

If Jack is selected as the Director, then he cannot be selected as the President, and thus, there are 4 choices (one of which is Jill) for the position of President. Thus, the company can fill the positions in 4 ways when Jack is the Director.

If Jill is selected as the Director, then she cannot be selected as the President, and thus, there are 4 choices (one of which is Jack) for the position of President. Thus, the company can fill the positions in 4 ways where Jill is the Director.

Suppose neither Jack nor Jill is selected for the position of Director. This means that one of the 5 - 2 = 3 candidates is chosen for this position. Further, since neither Jack nor Jill is selected as the Director, there are 5 candidates for the position of President (including Jack and Jill). Thus, there are 3 x 5 = 15 ways the company can fill the positions where neither Jack nor Jill is the Director.

In total, there are 4 + 4 + 15 = 23 ways the company can fill the positions.

Answer: B
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If Jack/Jill is not selected as D, then D has 3 choices, and P has 5 choices.
Therefore, no. of choices = 3*5

But is this not possible in 2 different ways? Like, if Jack/Jill is selected as D, then we have 3*5 choices. But the other way - if Jack/Jill is selected as P, we again have 3*5 choices. Right??

Its simple there are 5x5 ways of selecting candidates
Now out of 25 ways there are only two ways which are not possible - one is jack at both places & other is Jill at both places
therefore total posiible ways = 25-1-1=23