A city council consists of seven Democrats and seven Republicans. If a committee of six
people is selected, find the probability of selecting two Democrats and four Republicans.
(Type a fraction. Simplify your answer.)

Question

UpStudy AI · Accepted Answer

The total number of ways to form a committee of six people from 14 is

$$
\binom{14}{6}.
$$

The number of ways to choose exactly 2 Democrats out of 7 is

$$
\binom{7}{2},
$$

and the number of ways to choose exactly 4 Republicans out of 7 is

$$
\binom{7}{4}.
$$

Thus, the number of favorable committees is

$$
\binom{7}{2} \times \binom{7}{4}.
$$

The probability is given by

$$
\frac{\binom{7}{2} \times \binom{7}{4}}{\binom{14}{6}}.
$$

We compute each term:

1. Compute $$\binom{7}{2}$$:

$$
\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21.
$$

2. Compute $$\binom{7}{4}$$:
   
Since $$\binom{7}{4} = \binom{7}{3}$$,

$$
\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.
$$

3. Compute $$\binom{14}{6}$$:

$$
\binom{14}{6} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 3003.
$$

Now, substituting these values, we have

$$
\frac{21 \times 35}{3003} = \frac{735}{3003}.
$$

To simplify, notice that both the numerator and the denominator are divisible by 21:

$$
\frac{735 \div 21}{3003 \div 21} = \frac{35}{143}.
$$

Thus, the simplified probability is

$$
\frac{35}{143}.
$$
The probability of selecting two Democrats and four Republicans is $$\frac{35}{143}$$.

A city council consists of seven Democrats and seven Republicans. If a committee of six
people is selected, find the probability of selecting two Democrats and four Republicans.
(Type a fraction. Simplify your answer.)

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A city council consists of seven Democrats and seven Republicans. If a committee of six people is selected, find the probability of selecting two Democrats and four Republicans. (Type a fraction. Simplify your answer.)

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Latest Probability Questions

A city council consists of seven Democrats and seven Republicans. If a committee of six
people is selected, find the probability of selecting two Democrats and four Republicans.
(Type a fraction. Simplify your answer.)