Selecting a Committee
Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly.


c. What is the probability that the committee is composed of three Democrats and four Republicans?

Verified step by step guidance

1. Identify the total number of senators, which is the sum of Democrats and Republicans: \$55 + 45 = 100$ senators.
2. Determine the total number of ways to select a committee of 7 senators from 100 senators. This is given by the combination formula: \(\binom{100}{7}\).
3. Calculate the number of ways to select exactly 3 Democrats from the 55 Democrats. This is \(\binom{55}{3}\).
4. Calculate the number of ways to select exactly 4 Republicans from the 45 Republicans. This is \(\binom{45}{4}\).
5. Use the multiplication rule to find the number of favorable committees by multiplying the two combinations: \(\binom{55}{3} \times \binom{45}{4}\). Then, find the probability by dividing this product by the total number of committees: \(\frac{\binom{55}{3} \times \binom{45}{4}}{\binom{100}{7}}\).

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where order does not matter. In this problem, we use combinations to count the number of ways to choose a specific number of Democrats and Republicans from their respective groups without regard to order.

Probability of a Specific Event

The probability of a specific event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Here, the favorable outcomes are committees with exactly three Democrats and four Republicans, and the total outcomes are all possible committees of seven senators.

Multiplication Principle in Counting

The multiplication principle states that if one event can occur in m ways and another independent event can occur in n ways, then both events together can occur in m × n ways. Selecting Democrats and Republicans independently uses this principle to find the total favorable combinations.