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?

There are 240 ways to arrange themselves in line

Step-by-step 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

15

step-by-step explanation:

answer: option a.

step-by-step explanation:

you know that the quadrilateral opqr ( a four sided figure) is inscribed in circle. this means that all its vertices are on the circle.

then, by definition, the opposite angles of the quadrilateral are supplementary, this means that the sum of their measures is 180 degrees.

you can observe in the figure that the angles ∠opq   and the angle ∠orq   are opposite, therefore:

then, can be used to d to calculate the measure of ∠opq