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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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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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Out of 7 gents and 4 ladies a committee of 5 is to be formed. The numb...
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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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