A political scientist wants to conduct a research study on a president's approval rating. The researcher has obtained data that states that 45% of citizens are in favor of the president. The researcher wants to determine the probability that 6 out of the next 8 individuals in his community are in favor of the president.
What is the binomial coefficient of this study?
28
To calculate the binomial coefficient, we need to use the formula for combinations, also known as "n choose k". The formula is:
C(n, k) = n! / (k! * (n-k)!)
In this case, we want to calculate the probability of having 6 out of 8 individuals in favor of the president. So, n (the total number of individuals) is 8, and k (the number of individuals in favor of the president) is 6.
Let's substitute these values into the formula:
C(8, 6) = 8! / (6! * (8-6)!)
= 8! / (6! * 2!)
Now, let's calculate the factorials:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
2! = 2 * 1
Substituting these values:
C(8, 6) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (2 * 1))
Canceling out the common terms:
C(8, 6) = (8 * 7) / (2 * 1)
Calculating the value, we get:
C(8, 6) = 56 / 2
= 28
Therefore, the binomial coefficient for this study is 28.