Question

Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but cannot visit site \(X\), the most suspicious site, on two consecutive days. In how many different orders can the inspector visit these sites?

Short answer

Inspector can visit the sites in 90720 days.

Step-by-step solution

Definition of Concept

String: A Number String is a collection of related math problems designed to teach number-based strategies. It is a 10- to 15-minute routine that can be used during math class. Gather students in an area where the entire class meets to prepare an area for using the number string.

Find in how many different orders can the inspector visit these sites

Considering the given information:

Number of different sites\({\rm{ = 5}}\)

Number of visit per day\({\rm{ = 1}}\)

Using the following concept:

n In distinguishable objects in r distinguishable objects can be distributed in \({\rm{C(n + r - 1,r - 1)}}\) ways.

Each of the five different sites is inspected twice by the inspector, who visits one site per day. Inspector is not permitted to visit the site 'X' on consecutive days.

As a result, the total number of days,

\(\begin{array}{l}{\rm{ = (6!) \times C(5 + 5 - 1,5)}}\\{\rm{ = (6!) \times C(9,5)}}\\{\rm{ = 720 \times }}\frac{{{\rm{9!}}}}{{{\rm{5!4!}}}}\\{\rm{ = 720 \times }}\frac{{{\rm{9 \times 8 \times 7 \times 6}}}}{{{\rm{4 \times 3 \times 2}}}}\\{\rm{ = 720 \times 126}}\\{\rm{ = 90720 days }}\end{array}\)

Therefore, the required total number of days is\({\rm{90720}}\).