Unique Alpha Beta Coding Logic

The problem requires assigning unique two-symbol codes using symbols $\alpha$ and $\beta$ to four items: P, Q, R, and S.

Possible Two-Symbol Codes

Using two symbols ($\alpha$, $\beta$), there are $2 \times 2 = 4$ possible unique combinations:

• $\alpha\alpha$
• $\alpha\beta$
• $\beta\alpha$
• $\beta\beta$

Given Code Assignments

We are given the codes for two items:

• P is coded as $\alpha\alpha$.
• Q is coded as $\alpha\beta$.

Determining Codes for R and S

Since P and Q have been assigned $\alpha\alpha$ and $\alpha\beta$ respectively, the remaining unique codes available for R and S are $\beta\alpha$ and $\beta\beta$.

Therefore, R and S must be coded using these two remaining combinations. The two possible scenarios are:

1. R = $\beta\alpha$, S = $\beta\beta$
2. R = $\beta\beta$, S = $\beta\alpha$

Matching the Options

We need to find the option that lists codes for R and S corresponding to one of these valid scenarios.

• Option 1 ($\beta\alpha$, $\alpha\beta$): Invalid, $\alpha\beta$ is used by Q.
• Option 2 ($\beta\beta$, $\alpha\alpha$): Invalid, $\alpha\alpha$ is used by P.
• Option 3 ($\alpha\beta$, $\beta\beta$): Invalid, $\alpha\beta$ is used by Q.
• Option 4 ($\beta\alpha$, $\beta\beta$): Valid, uses the remaining unique codes.

Thus, R and S can be coded as $\beta\alpha$ and $\beta\beta$ respectively.