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?
How many ways to arrange 6 without restriction? 6!
In the total of ways, half of the time J will be behind F or F behind J.
\(=6!/2 = 360\)
Solution 2: To get a little more practice on permutations...
We could count the possibilities of J(...)F.
0 person in between: 5
1 person in between: 4
2 persons in between: 3
3 persons in between: 2
4 persons in between: 1
(5 + 4 + 3 + 2 + 1) * 4! = 360
Impossible is nothing to God.