QuestionJuly 17, 2025

Watch the video and then solve the problem given below. Click here to watch the video. A committee has seven men and four women. If four people are selected to go to a conference, what is the chance that the group is two men and two women? square (Type an integer or a simplified fraction.)

Watch the video and then solve the problem given below. Click here to watch the video. A committee has seven men and four women. If four people are selected to go to a conference, what is the chance that the group is two men and two women? square (Type an integer or a simplified fraction.)
Watch the video and then solve the problem given below.
Click here to watch the video.
A committee has seven men and four women.
If four people are selected to go to a conference, what is the chance that the group is two
men and two women?
square 
(Type an integer or a simplified fraction.)

Solution
4.0(244 votes)

Answer

\(\frac{7}{55}\) Explanation 1. Calculate Total Ways to Select 4 People Use combination formula: \binom{n}{r} = \frac{n!}{r!(n-r)!}. Total people = 11. Choose 4: \binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330. 2. Calculate Ways to Select 2 Men Men = 7. Choose 2: \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21. 3. Calculate Ways to Select 2 Women Women = 4. Choose 2: \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6. 4. Calculate Ways to Select 2 Men and 2 Women Multiply results from Steps 2 and 3: 21 \times 6 = 126. 5. Calculate Probability Divide favorable outcomes by total outcomes: \frac{126}{330} = \frac{7}{55}.

Explanation

1. Calculate Total Ways to Select 4 People<br /> Use combination formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. Total people = 11. Choose 4: $\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$.<br /><br />2. Calculate Ways to Select 2 Men<br /> Men = 7. Choose 2: $\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21$.<br /><br />3. Calculate Ways to Select 2 Women<br /> Women = 4. Choose 2: $\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$.<br /><br />4. Calculate Ways to Select 2 Men and 2 Women<br /> Multiply results from Steps 2 and 3: $21 \times 6 = 126$.<br /><br />5. Calculate Probability<br /> Divide favorable outcomes by total outcomes: $\frac{126}{330} = \frac{7}{55}$.
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