Problem Solving

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?

1) 6
2) 24
3) 120
4) 360
5) 720

another approach

Total - ( 1x5! + 4x1x4x3x2x1 + 4x3x1x3x2x1 + 4x3x2x1x2x1 + 4x3x2x1x1x1 )
720 - ( 120 + 96 + 72 + 48 + 24)
720 - 360 = 360

rishianand7 wrote: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?

1) 6
2) 24
3) 120
4) 360
5) 720

An alternate approach:

Let the 6 mobsters be A, B, C, D, F and J.
Direction of the line:
Front....Back.

Number of options for A = 6. (Any of the 6 positions.)
Number of options for B = 5. (Any of the 5 remaining positions.)
Number of options for C = 4. (Any of the 4 remaining positions.)
Number of options for D = 3. (Any of the 3 remaining positions.)

Of the two remaining positions, F must occupy the one more to the right, so that he can keep an eye on J.
Number of options for F = 1. (Of the two remaining positions, the one more to the right.)
Number of options for J = 1. (One position left.)

To combine these options, we multiply:
6*5*4*3*1*1 = 360.

The correct answer is D.