Question

a group consists of seven democrats and five republicans. four people are selected to attend a conference.
a. in how many ways can four people be selected from this group of twelve?
b. in how many ways can four republicans be selected from the five republicans?
c. find the probability that the selected group will consist of all republicans.

a. the number of ways to select four people from the group of twelve is \boxed{}

Explanation:

Step1: Identify the formula for combinations

The problem involves selecting a group of people where the order does not matter, so we use the combination formula \( C(n, k) = \frac{n!}{k!(n - k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to select. Here, \( n = 12 \) (total people) and \( k = 4 \) (people to select).

Step2: Substitute values into the formula

Substitute \( n = 12 \) and \( k = 4 \) into the combination formula:

$$ C(12, 4) = \frac{12!}{4!(12 - 4)!} = \frac{12!}{4! \cdot 8!} $$

Step3: Simplify the factorials

We know that \( n! = n \times (n - 1) \times \dots \times 1 \), so we can simplify the factorials:

$$ \frac{12!}{4! \cdot 8!} = \frac{12 \times 11 \times 10 \times 9 \times 8!}{4 \times 3 \times 2 \times 1 \times 8!} $$

The \( 8! \) terms cancel out, leaving:

$$ \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} $$

Step4: Calculate the numerator and denominator

Calculate the numerator: \( 12 \times 11 \times 10 \times 9 = 11880 \)
Calculate the denominator: \( 4 \times 3 \times 2 \times 1 = 24 \)

Step5: Divide the numerator by the denominator

$$ \frac{11880}{24} = 495 $$

Answer:

495