Understanding the Coding Pattern

The problem asks us to find the coding rule used for the letters 'Z', 'R', and 'J' and then apply this rule to find the code for the letter 'P'. We are given the following codes:

• 'Z' is coded as 261.
• 'R' is coded as 189.
• 'J' is coded as 1017.

Let's determine the standard alphabetical position for each letter (A=1, B=2, ..., Z=26).

• The position of 'Z' is 26.
• The position of 'R' is 18.
• The position of 'J' is 10.

Deriving the Coding Rule

Now, let's compare the alphabetical position with the given codes:

Observe the structure of the codes:

• For 'Z' (Position 26), the code is 261. It looks like the position (26) is followed by '1'.
• For 'R' (Position 18), the code is 189. It looks like the position (18) is followed by '9'.
• For 'J' (Position 10), the code is 1017. It looks like the position (10) is followed by '17'.

The numbers appended (1, 9, 17) need a clear connection to the letters or their positions. Let's consider the reverse alphabetical position (Z=1, Y=2, ..., A=26).

The formula for reverse alphabetical position is: $ReversePosition = 26 - N + 1$, where $N$ is the standard alphabetical position.

• For 'Z': $N=26$. Reverse Position = $26 - 26 + 1 = 1$.
• For 'R': $N=18$. Reverse Position = $26 - 18 + 1 = 9$.
• For 'J': $N=10$. Reverse Position = $26 - 10 + 1 = 17$.

Now, let's check if the code is formed by concatenating the standard alphabetical position ($N$) with its reverse alphabetical position:

• 'Z': Position = 26, Reverse Position = 1. Concatenated code = 261. (Matches)
• 'R': Position = 18, Reverse Position = 9. Concatenated code = 189. (Matches)
• 'J': Position = 10, Reverse Position = 17. Concatenated code = 1017. (Matches)

The coding rule is confirmed: The code for a letter is formed by writing its alphabetical position followed by its reverse alphabetical position.

Applying the Rule to Find the Code for 'P'

We need to find the code for the letter 'P'.

• Step 1: Find the standard alphabetical position of 'P'. 'P' is the 16th letter, so $N = 16$.
• Step 2: Find the reverse alphabetical position of 'P'. Reverse Position = $26 - N + 1 = 26 - 16 + 1 = 10 + 1 = 11$.
• Step 3: Concatenate the position and the reverse position. Code = Position || Reverse Position = 16 || 11 = 1611.

Therefore, 'P' will be coded as 1611.