Correct Option (B)
The problem requires determining the number of distinct combinations to select 3 persons from a row of 6, such that no two selected persons are consecutive.
First, calculate the total number of ways to choose 3 persons from 6 without any restrictions. This is determined using the combination formula:
6C3=3!(6−3)!6!=3!3!6!=3×2×1×3!6×5×4×3!=20
Next, identify and subtract the combinations where at least two selected persons are consecutive. These can be categorized into two cases:
Case 1: All three chosen persons are consecutive.
- (P1, P2, P3)
- (P2, P3, P4)
- (P3, P4, P5)
- (P4, P5, P6)
There are 4 such combinations.
Case 2: Exactly two chosen persons are consecutive.
This implies two persons form a consecutive pair, and the third person is not adjacent to either of them.
- If (P1, P2) are chosen, the third person cannot be P3. Possible choices: P4, P5, P6 (3 combinations).
- If (P2, P3) are chosen, the third person cannot be P1 or P4. Possible choices: P5, P6 (2 combinations).
- If (P3, P4) are chosen, the third person cannot be P2 or P5. Possible choices: P1, P6 (2 combinations).
- If (P4, P5) are chosen, the third person cannot be P3 or P6. Possible choices: P1, P2 (2 combinations).
- If (P5, P6) are chosen, the third person cannot be P4. Possible choices: P1, P2, P3 (3 combinations).
The total number of combinations where exactly two persons are consecutive is 3 + 2 + 2 + 2 + 3 = 12.
The total number of restricted combinations (where at least two persons are consecutive) = (Case 1) + (Case 2) = 4 + 12 = 16.
Therefore, the number of distinct possible combinations where no two persons are consecutive is calculated by subtracting the restricted combinations from the total combinations:
20 - 16 = 4.
The final answer is 4.
Incorrect Options:
Options A (3), C (5), and D (6) are incorrect. These values do not correspond to the precise calculation, which systematically accounts for all possible combinations and excludes those where selected persons are consecutive.