Permutation and Combination: In how many ways can 7 students be selected in a row of 7 seats if 2 of the students insist of sitting beside each other? In a gathering, the host make sure that each guest shakes hands with everyone else. If there are 30 guest how many handshakes will done?

Answers

1.

The 2 students insisting on sitting next to each other are considered as 1 unit.

In this case, we have 6 students who can be seated in 6! ways and the 2 students can be seated amongst themselves in 2! ways

Therefore, the required number of ways = 6!*2! = 1440 ways

2.

Number of guests: n = 30

Number of handshakes for each guest: n- 1

(Note that a guest does not shake his own hands)

Number of people involved in each handshake: 2

Total handshakes = [(n) (n-1)] / 2

= [(30)(30-1)] / 2

= [(30)(29)] / 2

= (870) / 2

= 435 handshakes