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 persons. In how many distinct possible combinations can the handshakes take place?

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:

$${{}_{}^{6}C}_3=\frac{6!}{3!(6-3)!}=\frac{6!}{3!3!}=\frac{6\times 5\times 4\times 3!}{3\times 2\times 1\times 3!}=20$$

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.