If the US Senate has 50 Democrats and 50 Republicans, what is the probability that a committee of 5 senators is comprised of all Democrats?

To solve this problem, we need to calculate the probability that a committee of 5 senators chosen from the Senate (which consists of 50 Democrats and 50 Republicans) is made up entirely of Democrats.

### Step 1: Total Number of Possible Committees
The total number of ways to choose 5 senators from the 100 senators (50 Democrats + 50 Republicans) is given by the combination formula:

\[
\text{Total number of committees} = \binom{100}{5}
\]

where \(\binom{n}{k}\) represents the number of ways to choose \(k\) objects from \(n\) objects, and is calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

So,

\[
\binom{100}{5} = \frac{100 \times 99 \times 98 \times 97 \times 96}{5 \times 4 \times 3 \times 2 \times 1} = 75287520
\]

### Step 2: Number of Ways to Choose All Democrats
Next, the number of ways to choose 5 Democrats from the 50 Democrats is:

\[
\text{Number of all-Democrat committees} = \binom{50}{5}
\]

This is:

\[
\binom{50}{5} = \frac{50 \times 49 \times 48 \times 47 \times 46}{5 \times 4 \times 3 \times 2 \times 1} = 2118760
\]

### Step 3: Probability of All Democrats Committee
The probability is the ratio of the number of all-Democrat committees to the total number of committees:

\[
\text{Probability} = \frac{\binom{50}{5}}{\binom{100}{5}} = \frac{2118760}{75287520} \approx 0.0281
\]

So, the probability that the committee of 5 senators is made up entirely of Democrats is approximately \(0.0281\), or 2.81%.

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Would you like more details on any part of the solution?

Here are 5 related questions you might explore:
1. What is the probability of selecting all Republicans instead of all Democrats?
2. How would the probability change if there were 51 Democrats and 49 Republicans?
3. What is the probability of selecting 3 Democrats and 2 Republicans?
4. How would the probability change if the committee size was increased to 6 senators?
5. What is the total number of ways to form a committee with at least 1 Democrat?

**Tip**: Understanding combinations is key to calculating probabilities when selection order doesn't matter.

Calculate the probability that a committee of 5 senators, chosen from an evenly split US Senate with 50 Democrats and 50 Republicans, is made up entirely of Democrats. This problem covers key concepts in probability and combinatorics, with detailed steps using the combination formula.

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