Subtopic: Permutations, Combinations & Probability
Question Statement:
P, Q, R and S are to be uniquely coded using α and β. If P is coded as αα and Q as αβ, then R and S, respectively, can be coded as
(A) βα and αβ
(B) ββ and αα
(C) αβ and ββ
(D) βα and ββ
Options:
(A) βα and αβ
(B) ββ and αα
(C) αβ and ββ
(D) βα and ββ
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2020 Q.7 / GA Q.7)**
**1. Binary Permutations Formulation**:
- Two symbols $\{\alpha, \beta\}$ used to construct distinct 2-character codewords for 4 elements:
$\text{Total 2-character permutations} = 2^2 = 4$
- The 4 unique codewords are:
$\{\alpha\alpha, \alpha\beta, \beta\alpha, \beta\beta\}$
**2. Unique Mapping**:
- Given:
- $P \to \alpha\alpha$
- $Q \to \alpha\beta$
- The remaining unused codewords are $\beta\alpha$ and $\beta\beta$.
- Assigning uniquely to $R$ and $S$:
$R = \beta\alpha, \quad S = \beta\beta$
- This matches **Option (D)**.
**3. Official Verification**:
The verified official IIT Delhi Master Answer is **Option (D)** (`βα and ββ`).
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.