Question

Six mobsters have arrived at the theater for the premiere of the film "Goodbuddies." One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand, though not necessarily right behind him. How many ways can the six arrange themselves in line such that Frankie’s requirement is satisfied?

Answer 1

There are 240 ways to arrange themselves in line

Explanation:

The number of ways that satisfied Frankie's requirement is defined by the position of Joey. Then we have the following cases:

1. Joey is the first in line: the number of ways for this case is calculated using a rule of multiplication as:

1 * 4 * 4 * 3 * 2 * 1 = 96

1st in Line 2nd 3rd 4th 5th 6th

Because, Joey is the first in line, then Frankie can not be right behind him, so we have 4 mobsters for be 2nd in line, and 4 mobsters for be 3rd in line, including Frankie, 3 mobsters for be 4th in Line, 2 mobsters for be 5th in line and 1 mobster for be 6th in line.

2. Joey is the second in line: the number of ways for this case is calculated using a rule of multiplication as:

4 * 1 * 3 * 3 * 2 * 1 = 72

1st in Line 2nd 3rd 4th 5th 6th

Because Joey is the second in line and Frankie can not be in neither the 1st or 3rd position of the line.

At the same way we can calculated the number of ways for the following cases:

3. Joey is the Third in line:

4 * 3 * 1 * 2 * 2 * 1 = 48

1st in Line 2nd 3rd 4th 5th 6th

4. Joey is the fourth in line:

4 * 3 * 2 * 1 * 1 * 1 = 24

1st in Line 2nd 3rd 4th 5th 6th

On the other hand, if Joey is 5th or 6th in line, the requirement of Frankie is not satisfy. Then, the total number of ways in which Frankie is satisfied is calculated as the sum of the cases mentioned above. This is:

96 + 72 + 48 + 24 = 240 ways

So, there are 240 ways to arrange themselves in line