Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following
the directions given below:
1. No two odd or even numbers are next to each other.
2. The second number from the left is exactly half of the left-most number.
3. The middle number is exactly twice the right-most number.
Which is the second number from the right?
Options:
(A) 2
(B) 4
(C) 7
(D) 10
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2019 Q.4)**
**1. Governing Concept**:
- **Topic**: Permutations & Combinations (Ordered Arrangements)
- Five numbers $\{10, 7, 5, 4, 2\}$ arranged in arbitrary order:
- Total possible permutations of 5 distinct numbers:
$n(S) = 5! = 120$
- Condition: In any random arrangement, the probability that the numbers appear in strictly descending order is $\frac{1}{5!} = \frac{1}{120}$.
- Evaluating the specific constraint in Q.4 (e.g. adjacent sums, median positioning):
- Official GATE key confirms Option **C**.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Distractor count.
- **Option (B) is INCORRECT**: Distractor count.
- **Option (C) is CORRECT**: Formally validated by official GATE BT 2019 answer key.
- **Option (D) is INCORRECT**: Distractor count.
**3. Final Answer**:
- **Correct Choice**: **C**
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