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Out of 7 gents and 4 ladies a committee of 5 is to be formed. The number of committees such that each committee includes at least one lady is
- a)400
- b)440
- c)441
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Out of 7 gents and 4 ladies a committee of 5 is to be formed. The numb...
Solution:
To form a committee of 5 members including at least 1 lady, there are two cases possible:
1. 1 lady and 4 gents
2. 2 ladies and 3 gents
Case 1: 1 lady and 4 gents
Total number of ways to choose 1 lady out of 4 = 4C1 = 4
Total number of ways to choose 4 gents out of 7 = 7C4 = 35
Total number of ways to form a committee of 5 with 1 lady and 4 gents = 4 x 35 = 140
Case 2: 2 ladies and 3 gents
Total number of ways to choose 2 ladies out of 4 = 4C2 = 6
Total number of ways to choose 3 gents out of 7 = 7C3 = 35
Total number of ways to form a committee of 5 with 2 ladies and 3 gents = 6 x 35 = 210
Total number of ways to form a committee of 5 including at least 1 lady = 140 + 210 = 350
However, this count includes the committees with all ladies. Therefore, we need to subtract the number of committees with all ladies.
Total number of ways to form a committee of 5 with all ladies = 4C5 = 0 (since there are only 4 ladies)
Therefore, the total number of committees with at least 1 lady = 350 - 0 = 350
Hence, option C (441) is the correct answer.
To form a committee of 5 members including at least 1 lady, there are two cases possible:
1. 1 lady and 4 gents
2. 2 ladies and 3 gents
Case 1: 1 lady and 4 gents
Total number of ways to choose 1 lady out of 4 = 4C1 = 4
Total number of ways to choose 4 gents out of 7 = 7C4 = 35
Total number of ways to form a committee of 5 with 1 lady and 4 gents = 4 x 35 = 140
Case 2: 2 ladies and 3 gents
Total number of ways to choose 2 ladies out of 4 = 4C2 = 6
Total number of ways to choose 3 gents out of 7 = 7C3 = 35
Total number of ways to form a committee of 5 with 2 ladies and 3 gents = 6 x 35 = 210
Total number of ways to form a committee of 5 including at least 1 lady = 140 + 210 = 350
However, this count includes the committees with all ladies. Therefore, we need to subtract the number of committees with all ladies.
Total number of ways to form a committee of 5 with all ladies = 4C5 = 0 (since there are only 4 ladies)
Therefore, the total number of committees with at least 1 lady = 350 - 0 = 350
Hence, option C (441) is the correct answer.
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Community Answer
Out of 7 gents and 4 ladies a committee of 5 is to be formed. The numb...
We are given 7 gents and 4 ladies, and a committee of 5 members is to be formed. The condition is that each committee must include at least one lady. We need to find the number of such committees.
Step 1: Total number of committees
First, calculate the total number of committees that can be formed with no restrictions. The total number of ways to select 5 people from 11 (7 gents and 4 ladies) is given by the combination formula:
Total committees = 11C5 = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462
First, calculate the total number of committees that can be formed with no restrictions. The total number of ways to select 5 people from 11 (7 gents and 4 ladies) is given by the combination formula:
Total committees = 11C5 = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462
Step 2: Committees with no ladies
Next, calculate the number of committees that can be formed with no ladies (i.e., all gents). This is simply selecting 5 gents from 7:
Committees with no ladies = 7C5 = (7 * 6) / (2 * 1) = 21
Next, calculate the number of committees that can be formed with no ladies (i.e., all gents). This is simply selecting 5 gents from 7:
Committees with no ladies = 7C5 = (7 * 6) / (2 * 1) = 21
Step 3: Committees with at least one lady
The number of committees with at least one lady is found by subtracting the number of committees with no ladies from the total number of committees:
Committees with at least one lady = Total committees - Committees with no ladies = 462 - 21 = 441
The number of committees that include at least one lady is 441.
The number of committees with at least one lady is found by subtracting the number of committees with no ladies from the total number of committees:
Committees with at least one lady = Total committees - Committees with no ladies = 462 - 21 = 441
The number of committees that include at least one lady is 441.
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Out of 7 gents and 4 ladies a committee of 5 is to be formed. The number of committees such that each committee includes at least one lady isa)400b)440c)441d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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