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From a group of 7 men and 6 women, five persons are to be selected to form a committee so that atleast 3 men are there on the committee. In how many ways can it be done ?
  • a)
    645
  • b)
    564
  • c)
    735
  • d)
    756
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
From a group of 7 men and 6 women, five persons are to be selected to ...
Given: 
(7 men + 6 women) 5 persons are to be chosen for a committee.
Formula used: nCr = n!/(n - r)! r!
Calculation:
Ways in which at least 3 men are selected;
⇒ 3 men + 2 women
⇒ 4 men + 1 woman 
⇒ 5 men + 0 woman 
Number of ways = 7C3 × 6C2 + 7C4 × 6C1 + 7C5 × 6C0
⇒ 7!/(3! × 4!) × 6!/(2! × 4!) + 7!/(4! × 3!) × 6!/(1! × 5!) + 7!/(5! × 2!) × 6!/(6!× 0!)
⇒ 35 × 15 + 35 × 6 + 21 
⇒ 735 + 21 = 756
∴ The required no of ways = 756.
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From a group of 7 men and 6 women, five persons are to be selected to ...
Problem Overview
We need to select a committee of 5 persons from a group of 7 men and 6 women, ensuring that at least 3 men are included.
Possible Combinations
To satisfy the condition of having at least 3 men, we have three scenarios:
  • Scenario 1: 3 men and 2 women
  • Scenario 2: 4 men and 1 woman
  • Scenario 3: 5 men

Calculating the Combinations
1. Scenario 1: 3 Men and 2 Women
- Choose 3 men from 7: C(7, 3)
- Choose 2 women from 6: C(6, 2)
- Total ways: C(7, 3) * C(6, 2)
2. Scenario 2: 4 Men and 1 Woman
- Choose 4 men from 7: C(7, 4)
- Choose 1 woman from 6: C(6, 1)
- Total ways: C(7, 4) * C(6, 1)
3. Scenario 3: 5 Men
- Choose 5 men from 7: C(7, 5)
- Total ways: C(7, 5)
Calculating Each Scenario
- C(7, 3) = 35, C(6, 2) = 15, thus Scenario 1 = 35 * 15 = 525
- C(7, 4) = 35, C(6, 1) = 6, thus Scenario 2 = 35 * 6 = 210
- C(7, 5) = 21, thus Scenario 3 = 21
Total Combinations
Total ways = Scenario 1 + Scenario 2 + Scenario 3
Total ways = 525 + 210 + 21 = 756
Final Answer
Thus, the total number of ways to form the committee is 756, which corresponds to option 'D'.
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In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girlsG1, G2, G3, G4, G5.(i) Let α1 be the total number of ways in which the committee can be formed such that the committee has5 members, having exactly 3 boys and 2 girls.(ii) Let α2 be the total number of ways in which the committee can be formed such that the committee hasat least 2 members, and having an equal number of boys and girls.(iii) Let α3be the total number of ways in which the committee can be formed such that the committee has5 members, at least 2 of them being girls.(iv) Let α4be the total number of ways in which the committee can be formed such that the committee has4 members, having atleast 2 girls and such that both M1 and G1 are NOT in the committee together.The correct option is

In a high school, a committee has to be formed from a group of 6 boys M1, M2,M3, M4,M5, M6 and 5 girls G1,G2, G3, G4, G5.(i) Let α1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.(ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.(iii) Let α3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.(iv) Let α4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both M1 and G1 are NOT in the committee together.The correct option is

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From a group of 7 men and 6 women, five persons are to be selected to form a committee so that atleast 3 men are there on the committee. In how many ways can it be done ?a)645b)564c)735d)756Correct answer is option 'D'. Can you explain this answer?
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