There are 6 persons arranged in a row. Another person has to shake hands with 3 of them so that he should not shake hands with two consecutive…

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:

$${_{}^{6}C}_{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4 \times 3!}{3 \times 2 \times 1 \times 3!} = 20$$6​C3​=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.