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 ?

• Label the 6 persons in the row as 1, 2, 3, 4, 5, and 6.

• The condition requires selecting 3 persons such that no two selected persons are adjacent (consecutive).

• If person 1 is selected, person 2 cannot be selected. The remaining 2 must be chosen from {3, 4, 5, 6} with no adjacency. The valid pairs are (3,5), (3,6), and (4,6). This gives combinations: {1,3,5}, {1,3,6}, {1,4,6}.

• If person 1 is not selected, the first selected person must be 2 or higher. If person 2 is selected, person 1 and 3 are excluded. The remaining 2 must be from {4, 5, 6} with no adjacency. The only valid pair is (4,6). This gives combination: {2,4,6}.

• If person 2 is not selected, the first selected person must be 3 or higher. If person 3 is selected, person 2 and 4 are excluded. The remaining 2 must be from {5, 6}, but they are consecutive, so no valid pair exists.

• Thus, the total distinct combinations are {1,3,5}, {1,3,6}, {1,4,6}, and {2,4,6}, which sums to 4.