GATE 2020 IN GA Q.7: solution and answer

Official answer: DSource: Official answer key, GATE 2020 IN (IIT Delhi)

1. The codes for P and Q have two symbols each, and every option also uses two-symbol codes. So we work with two-symbol strings made from $\alpha$ and $\beta$.

2. By the multiplication principle there are 2 choices for each of the 2 positions, so $2\times 2=4$ such codes: $\alpha \alpha ,\alpha \beta ,\beta \alpha ,\beta \beta$. That is exactly one code for each of the four letters.

3. "Uniquely coded" means no two letters can share a code. P already has $\alpha \alpha$ and Q already has $\alpha \beta$, so only $\beta \alpha$ and $\beta \beta$ are left for R and S.

4. Check each option. (A) gives S the code $\alpha \beta$, which Q already has. (B) gives S the code $\alpha \alpha$, which P already has. (C) gives R the code $\alpha \beta$, which Q already has. All three break uniqueness, so they are rejected.

5. (D) gives R $=\beta \alpha$ and S $=\beta \beta$. Neither code is used by P or Q, and the two are different from each other, so all four letters get different codes. The answer is (D).