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07 Dec 2024, 10:38
Kudos
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
86% (02:40) correct
14%
(02:15)
wrong
based on 65
sessions
History
Date
Time
Result
Not Attempted Yet
A team of 6 members is to be selected from a group of 7 men and 5 women. How many different teams can be formed if the team must include at least 3 women?
A. 540
B. 580
C. 612
D. 700
E. 462
A. 540
B. 580
C. 612
D. 700
E. 462
07 Dec 2024, 17:44
Kudos
Bookmarks
Three options
3W, 3M - 5C3 x 7C3 = 350
4W, 2M - 5C4 x 7C2 = 105
5W, 1M - 1 x 7 = 7
Total = 462
3W, 3M - 5C3 x 7C3 = 350
4W, 2M - 5C4 x 7C2 = 105
5W, 1M - 1 x 7 = 7
Total = 462
08 Dec 2024, 09:06
Kudos
Bookmarks
Problem:
A team of 6 members is to be selected from a group of 7 men and 5 women. How many different teams can be formed if the team must include at least 3 women?
Answer Choices:
A. 540
B. 580
C. 612
D. 700
E. 462
Solution:
To determine the number of different teams that can be formed with at least 3 women, we calculate the possibilities for each case and sum them.
Case 1: 3 Women and 3 Men
- Number of ways to choose 3 women from 5:
\(\)
\[
5C3 = \frac{5!}{3!(5-3)!} = 10
\]
- Number of ways to choose 3 men from 7:
\[
7C3 = \frac{7!}{3!(7-3)!} = 35
\]
- Total teams for this case:
\[
10 \times 35 = 350
\]
Case 2: 4 Women and 2 Men
- Number of ways to choose 4 women from 5:
\[
5C4 = \frac{5!}{4!(5-4)!} = 5
\]
- Number of ways to choose 2 men from 7:
\[
7C2 = \frac{7!}{2!(7-2)!} = 21
\]
- Total teams for this case:
\[
5 \times 21 = 105
\]
Case 3: 5 Women and 1 Man
- Number of ways to choose 5 women from 5:
\[
5C5 = \frac{5!}{5!(5-5)!} = 1
\]
- Number of ways to choose 1 man from 7:
\[
7C1 = \frac{7!}{1!(7-1)!} = 7
\]
- Total teams for this case:
\[
1 \times 7 = 7
\]
Total Number of Teams:
Summing the results from all cases:
\[
350 + 105 + 7 = 462
\]
Final Answer:
E. 462
A team of 6 members is to be selected from a group of 7 men and 5 women. How many different teams can be formed if the team must include at least 3 women?
Answer Choices:
A. 540
B. 580
C. 612
D. 700
E. 462
Solution:
To determine the number of different teams that can be formed with at least 3 women, we calculate the possibilities for each case and sum them.
Case 1: 3 Women and 3 Men
- Number of ways to choose 3 women from 5:
\(\)
\[
5C3 = \frac{5!}{3!(5-3)!} = 10
\]
- Number of ways to choose 3 men from 7:
\[
7C3 = \frac{7!}{3!(7-3)!} = 35
\]
- Total teams for this case:
\[
10 \times 35 = 350
\]
Case 2: 4 Women and 2 Men
- Number of ways to choose 4 women from 5:
\[
5C4 = \frac{5!}{4!(5-4)!} = 5
\]
- Number of ways to choose 2 men from 7:
\[
7C2 = \frac{7!}{2!(7-2)!} = 21
\]
- Total teams for this case:
\[
5 \times 21 = 105
\]
Case 3: 5 Women and 1 Man
- Number of ways to choose 5 women from 5:
\[
5C5 = \frac{5!}{5!(5-5)!} = 1
\]
- Number of ways to choose 1 man from 7:
\[
7C1 = \frac{7!}{1!(7-1)!} = 7
\]
- Total teams for this case:
\[
1 \times 7 = 7
\]
Total Number of Teams:
Summing the results from all cases:
\[
350 + 105 + 7 = 462
\]
Final Answer:
E. 462
