Q30. - Algebra 2

Question

The manager of a four-screen movie theater is deciding which of 12 available movies to show. The screens are in rooms with different seating capacities. How many ways can he show four different movies on the screen.

Step-by-Step Solution

Answer

The number of ways the manager can show four different movies on the screen is 11880. 

Step 1. Given Information.

Given the manager of a four-screen movie theater is deciding which of 12 available movies to show. The screens are in rooms with different seating capacities. The number of ways he can show four different movies on the screen is to be determined.

Step 2. Calculation .

A permutation is when n objects are available and r are to be picked and arranged in a certain order and the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

A combination is when n objects are available and r are to be picked without arrangement and the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Here, since the screens are different the order of choosing does affect the final outcome i.e., every arrangement is different. Hence the given situation is a permutation.

 

Here n=12,r=4

Plugging the values:

 $\begin{array}{l}P\left(12,4\right)=\frac{12!}{\left(12-4\right)!}\\ P\left(12,4\right)=\frac{12!}{8!}\\ P\left(12,4\right)=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8!}{8!}\\ P\left(12,4\right)=12\cdot 11\cdot 10\cdot 9\\ P\left(12,4\right)=11880\end{array}$

Step 3. Conclusion .

Hence, the number of ways the manager can show four different movies on the screen is 11880.