The problem asks us to find the number of distinct seating arrangements for four persons P, Q, R, and S in a row, with a specific condition: R should not be seated at the second position from the left end of the row.

Seating Arrangements: Total Possibilities

First, let us determine the total number of ways to arrange four distinct persons (P, Q, R, S) in four positions in a row without any restrictions. This is a classic permutation problem. For 'n' distinct items to be arranged in 'n' positions, the number of permutations is given by $n!$ (n factorial).

• Number of persons ($n$) = 4
• Total positions = 4
• Total arrangements = $4!$

Let's calculate the factorial:

$\qquad 4! = 4 \times 3 \times 2 \times 1 = 24$

So, there are 24 distinct ways to seat P, Q, R, and S in a row without any specific conditions.

Constraint Analysis: R's Position

The problem states a constraint: R should not be seated at the second position from the left end of the row. To solve this, it's often easier to find the number of arrangements where the condition is violated (i.e., R is seated at the second position), and then subtract this from the total number of arrangements.

Arrangements Where R is at the Second Position

Let's consider the scenario where R is fixed at the second position from the left. The positions are represented as follows:

_ _ _ _

If R is at the second position, it looks like this:

_ R _ _

Now, we have 3 remaining persons (P, Q, S) and 3 remaining positions (1st, 3rd, and 4th). The number of ways to arrange these 3 persons in the 3 remaining positions is $3!$ (3 factorial).

• Remaining persons = P, Q, S (3 persons)
• Remaining positions = 1st, 3rd, 4th (3 positions)
• Arrangements with R fixed at 2nd position = $3!$

Let's calculate this factorial:

$\qquad 3! = 3 \times 2 \times 1 = 6$

So, there are 6 arrangements where person R is seated at the second position from the left.

Calculating Distinct Seating Arrangements with the Constraint

To find the number of distinct seating arrangements where R is not seated at the second position, we subtract the "unwanted" arrangements (where R is at the second position) from the total possible arrangements.

• Total arrangements without restrictions = 24
• Arrangements where R is at the second position = 6

Number of distinct seating arrangements satisfying the condition = (Total arrangements) - (Arrangements where R is at the second position)

$\qquad \text{Distinct arrangements} = 24 - 6 = 18$

Therefore, there are 18 distinct seating arrangements possible where R is not seated at the second position from the left end of the row.

Summary of Seating Calculation

In summary, by using the principles of permutations and addressing the specific constraint, we found that the total number of distinct ways to seat the four persons P, Q, R, and S, such that R is not at the second position, is 18.