(S1): the average of 4 numbers is 25

(S2): Each number is at most 40

(S3): Each number is at least 20.

Choose the option that is necessarily correct.

The correct option is S1 & S3 together imply S2.

To understand why this statement is necessarily correct, let us analyze the given statements mathematically.
Let the four numbers be represented as $x_1$, $x_2$, $x_3$, and $x_4$.

1. Analyzing Statement (S1):
The average of the 4 numbers is 25. The mathematical expression for the average is:

$\frac{x_1+x_2+x_3+x_4}{4}=25$

Multiplying both sides by 4 gives us the sum of the four numbers:

$x_1+x_2+x_3+x_4=100$

2. Analyzing Statement (S3):
Each number is at least 20. This gives us a lower bound for each individual number:

$x_i\ge 20$

for all $i\in \{1,2,3,4\}$.

3. Determining the Maximum Possible Value of Any Number (Proving S2):
Let us find the maximum possible value that any single number, say $x_1$, can take. We can express $x_1$ in terms of the other three numbers using the sum equation from (S1):

$x_1=100-(x_2+x_3+x_4)$

To make $x_1$ as large as possible, we must minimize the sum of the remaining three numbers ($x_2+x_3+x_4$).
Since statement (S3) tells us that each number must be at least 20, the absolute minimum value for each of these three numbers is 20. Thus, their minimum combined sum is:

$x_2+x_3+x_4\ge 20+20+20=60$

Substituting this minimum sum back into the equation for $x_1$:

$x_1\le 100-60$

$x_1\le 40$

Since $x_1$ was chosen arbitrarily, this mathematical limit applies to all four numbers. Therefore, each of the numbers is at most 40. This is exactly statement (S2). Thus, statements (S1) and (S3) together logically guarantee that statement (S2) must be true.