While attending a film festival you decide that there are 12 movies that you are interested in seeing. However, you only have time to see 7 movies. In how many different ways can you watch 8 of the 12 movies?
A. 19,958,400
B. 12
C. 9,979,200***
D. 1,663,200
I thought you only had time to watch 7 movies, so which is it ?
btw, P(12,8) = 19,958,400
To calculate the number of different ways you can watch 8 of the 12 movies, you can use the combination formula.
The number of combinations of selecting r items from a set of n items is given by:
nCr = n! / ((n-r)! * r!)
In this case, you want to select 8 movies out of 12, so n = 12 and r = 8.
Plugging the values into the formula:
12C8 = 12! / ((12-8)! * 8!)
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
(12-8)! = 4!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Calculating the value:
12C8 = (12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
12C8 = 12 * 11 * 10 * 9 / (4 * 3 * 2 * 1)
12C8 = 9,979,200
Therefore, the correct answer is C. 9,979,200.
To solve this problem, we can use the concept of combinations. The number of ways we can choose 8 movies from a total of 12 movies is given by the binomial coefficient C(12, 8).
The formula for the binomial coefficient is C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number we want to choose.
In this case, we have n = 12 (total number of movies) and r = 8 (number of movies we want to choose).
Using the formula, we can calculate:
C(12, 8) = 12! / (8! * (12-8)!)
Simplifying the expression, we get:
C(12, 8) = 12! / (8! * 4!)
Now, let's calculate the factorials:
12! = 12 * 11 * 10 * 9 * 8!
8! = 8 * 7 * 6 * 5 * 4!
4! = 4 * 3 * 2 * 1
Substituting the factorials back into the expression, we get:
C(12, 8) = (12 * 11 * 10 * 9 * 8!) / (8 * 7 * 6 * 5 * 4!)
The factorials in the numerator and denominator cancel out, leaving us with:
C(12, 8) = (12 * 11 * 10 * 9) / (8 * 7 * 6 * 5)
Now, let's calculate the values:
C(12, 8) = 9,979,200 / 1680
Simplifying this division, we get:
C(12, 8) = 5,940
Therefore, there are 5,940 different ways to watch 8 movies from a total of 12 movies. However, none of the given answer options match this value. Therefore, there seems to be an error in the answer choices.