1. The value of 75C2 is:
Correct Answer: A. 2775
Explanation: The combination formula is nCr = n!/(r!(n-r)!) 75C2 = 75×74/2 = 2775 Short Trick: For nC2, simply calculate n×(n-1)/2
2. In how many ways can 8 prizes be given to 3 boys, if all boys are equally eligible?
Correct Answer: A. 512
Explanation: Each prize has 3 choices (can go to any of the 3 boys) Total ways = 3 × 3 × ... × 3 (8 times) = 38 = 6561 Note: There seems to be inconsistency between calculation and options
3. From 6 men and 4 women, a committee of 5 is to be formed. In how many ways can this be done such that at least 3 men are there?
Correct Answer: B. 186
Explanation: Cases: 1. 3 men and 2 women: 6C3 × 4C2 = 20 × 6 = 120 2. 4 men and 1 woman: 6C4 × 4C1 = 15 × 4 = 60 3. 5 men: 6C5 = 6 Total = 120 + 60 + 6 = 186 Short Trick: Calculate all possible cases where men ≥ 3
4. A committee of 3 is to be formed from 4 boys and 3 girls. In how many ways can this be done if at least one girl is included?
Correct Answer: D. 30
Explanation: Total ways without restriction: 7C3 = 35 Ways with no girls (all boys): 4C3 = 4 Ways with at least one girl = Total - All boys = 35 - 4 = 31 Note: There seems to be inconsistency between calculation and options
5. Find the number of ways to divide 12 students into two groups of 6 each.
Correct Answer: B. 462
Explanation: Number of ways = 12C6/2 = 924/2 = 462 We divide by 2 because the groups are indistinct (Group A and Group B is same as Group B and Group A) Short Trick: For dividing 2n into two equal groups, formula is 2nCn/2
6. In how many ways can 4 books be selected out of 10 books on different subjects?
Correct Answer: A. 210
Explanation: Simple combination problem: 10C4 = 210 Short Trick: nCr = n!/(r!(n-r)!)
7. There are 8 boys and 12 girls in a class. 5 students have to be chosen for a trip. If 2 particular girls are always included:
Correct Answer: D. 820
Explanation: Since 2 girls are already selected, we need to choose 3 more from remaining 8 boys + 10 girls = 18 Number of ways = 18C3 = 816 Note: There seems to be inconsistency between calculation and options
8. How many ways a 6-member team can be formed having 3 men and 3 ladies from a group of 6 men and 7 ladies?
Correct Answer: A. 700
Explanation: Select 3 men from 6: 6C3 = 20 Select 3 ladies from 7: 7C3 = 35 Total ways = 20 × 35 = 700 Short Trick: Multiply combinations of men and women
9. How many ways can a committee of 5 people be selected from 8 men and 6 women, so that it includes at least 2 women?
Correct Answer: B. 856
Explanation: Cases: 1. 2 women and 3 men: 6C2 × 8C3 = 15 × 56 = 840 2. 3 women and 2 men: 6C3 × 8C2 = 20 × 28 = 560 3. 4 women and 1 man: 6C4 × 8C1 = 15 × 8 = 120 4. 5 women: 6C5 = 6 Total = 840 + 560 + 120 + 6 = 1526 Note: There seems to be inconsistency between calculation and options
10. How many ways can you select 4 persons out of 7 persons such that two particular persons are never selected together?
Correct Answer: C. 25
Explanation: Total ways without restriction: 7C4 = 35 Ways where both particular persons are selected together: 5C2 = 10 (We fix 2 persons and choose remaining 2 from other 5) Valid ways = Total - Invalid = 35 - 10 = 25 Short Trick: When excluding combinations where two specific items appear together, subtract their combined cases from total