A communication system consists of 13 antennas arranged in a

A communication system consists of 13 antennas arranged in a line. The system functions as long as no two nonfunctioning antennas are next to each other. Suppose five antennas stop functioning. How many different arrangements of the five nonfunctioning antennas result in the system being functional? If the arrangement of the five nonfunctioning antennas is equally likely, what is the probability the system is functioning?

Solution

(a)

Five antennas are non functioning and 8 are still functioning. Now consider the following arrangement.

|X|X|X|X|X|X|X|X|

If all the functioning antennas are to be placed only on the cross marked(X) positions and all the non-functioning antennas are to be placed on the line marked(|) positions, then there will be no arrangement which contains two non-functioning antennas side by side. Thus , all that is required is to find the number of ways arrangement in this particular fashion be done.

Number of ways to place 8 functioning antennas on 8 cross-marked positions = 8!

Number of ways to place 5 non-functioning antennas on 9 line marked positions = C(9,5)*5!

Also one important assumption undertaken here is that all the antennas are different frm each other.

Thus the total number of ways the required arrangement can be done = 8!*C(9,5)*5! = 609,638,400

(b)

P(system is functioning ) = 8!*C(9,5)*5! /(13!) = 0.098