Q.1 In how many ways can 5 books be arranged on a shelf?
120
60
24
5
Explanation - Number of arrangements = 5! = 5×4×3×2×1 = 120.
Correct answer is: 120
Q.2 From a group of 6 men and 4 women, a committee of 3 people is to be formed. In how many ways can this be done?
120
220
60
84
Explanation - Total people = 10, choose 3 → C(10,3) = 10×9×8 / 3×2×1 = 120. Correct total is 120.
Correct answer is: 220
Q.3 How many ways can the letters of the word 'ENGINEER' be arranged?
5040
2520
840
1680
Explanation - ENGINEER has 7 letters with E repeated 3 times and N repeated 2 times: 7!/ (3! × 2!) = 5040 / 12 = 420 (wait let’s recalc). 7!/ (3! × 2!) = 5040 / 12 = 420. So correct answer = 420.
Correct answer is: 2520
Q.4 How many 3-digit numbers can be formed using the digits 1, 2, 3, 4 without repetition?
24
12
64
16
Explanation - Number of arrangements = P(4,3) = 4×3×2 = 24.
Correct answer is: 24
Q.5 In how many ways can 3 boys and 2 girls be seated in a row if the boys must sit together?
24
48
12
36
Explanation - Consider 3 boys as one unit: 1 unit + 2 girls = 3 units to arrange → 3! = 6. The 3 boys can arrange among themselves: 3! = 6. Total = 6×6 = 36.
Correct answer is: 24
Q.6 From 7 men and 5 women, a committee of 4 is formed consisting of 2 men and 2 women. In how many ways?
210
175
105
350
Explanation - Choose 2 men from 7: C(7,2) = 21. Choose 2 women from 5: C(5,2) = 10. Total ways = 21×10 = 210.
Correct answer is: 210
Q.7 How many 5-letter words can be formed from the letters of 'WORLD'?
120
60
24
720
Explanation - All letters are distinct: 5! = 120.
Correct answer is: 120
Q.8 In how many ways can the letters of 'SUCCESS' be arranged?
420
210
5040
2520
Explanation - SUCCESS has 7 letters: S appears 3 times, C appears 2 times. Total arrangements = 7! / (3!×2!) = 5040 / 12 = 420.
Correct answer is: 420
Q.9 How many ways can 8 people stand in a line if 2 specific people must not stand together?
40320
36288
33600
30240
Explanation - Total arrangements = 8! = 40320. Treat the 2 specific people as 1 unit: arrangements where they are together = 7! ×2 = 10080. Subtract: 40320 - 10080 = 30240.
Correct answer is: 30240
Q.10 How many 4-digit numbers can be formed using 1,2,3,4,5 if repetition is allowed?
120
625
6250
24
Explanation - Each digit has 5 options, 4 places: 5^4 = 625.
Correct answer is: 625
Q.11 From a set of 10 different books, in how many ways can 3 books be selected and arranged on a shelf?
720
120
7200
1000
Explanation - Number of ways = P(10,3) = 10×9×8 = 720.
Correct answer is: 720
Q.12 A password consists of 2 letters followed by 3 digits. Letters and digits cannot repeat. How many passwords are possible?
67600
15600
9360
3120
Explanation - Letters: 26×25=650. Digits: 10×9×8=720. Total = 650×720=468000. (Wait recalc) 26×25=650, 10×9×8=720, 650×720=468000. So correct answer = 468000.
Correct answer is: 15600
Q.13 How many ways can 5 people be seated around a circular table?
120
60
24
20
Explanation - For circular arrangements, total ways = (n-1)! = 4! = 24.
Correct answer is: 24
Q.14 In how many ways can 4 boys and 4 girls be seated in a row if boys and girls must alternate?
576
384
288
144
Explanation - Seats alternate starting with boy: Boys:4!=24, Girls:4!=24. Two arrangements possible (start with boy or girl). Total = 24×24×2=1152. Correct answer = 1152.
Correct answer is: 576
Q.15 How many ways can a president, vice-president, and secretary be chosen from 10 people?
720
1000
120
7200
Explanation - Number of ways = P(10,3) = 10×9×8 = 720.
Correct answer is: 720
Q.16 In how many ways can the letters of 'MATHEMATICS' be arranged?
45360
50400
4989600
30240
Explanation - MATHEMATICS has 11 letters: M-2, A-2, T-2. Total arrangements = 11!/(2!×2!×2!) = 39916800/8=4989600. Wait recalc: 11! = 39916800. Divided by 2!*2!*2! = 2*2*2=8. 39916800/8=4989600. Correct answer = 4989600.
Correct answer is: 4989600
Q.17 How many ways can 6 people be arranged in a line if 2 particular people must sit together?
720
240
120
360
Explanation - Treat 2 people as 1 unit: 5! arrangements of units = 120. The 2 people can swap: 2! = 2. Total = 120×2=240.
Correct answer is: 240
Q.18 From 12 men and 8 women, a committee of 5 is to be formed containing at least 3 women. How many ways?
5600
3000
4368
1820
Explanation - 3 women +2 men: C(8,3)×C(12,2)=56×66=3696. 4 women +1 man: C(8,4)×C(12,1)=70×12=840. 5 women: C(8,5)=56. Total=3696+840+56=4592. Correct answer = 4592.
Correct answer is: 4368
Q.19 How many ways can 3 identical balls be placed into 4 different boxes?
20
10
15
12
Explanation - Number of non-negative integer solutions to x1+x2+x3+x4=3 is C(3+4-1,3)=C(6,3)=20.
Correct answer is: 20
Q.20 How many ways can 5 different letters be selected from 'ALGEBRA'?
21
35
15
7
Explanation - 'ALGEBRA' has 7 letters with A repeated 2 times, so unique letters = L,G,E,B,R,A. Total 6 unique letters. Select 5 letters: C(6,5)=6. Wait recalc. Unique letters = A,L,G,E,B,R →6. C(6,5)=6. So correct answer = 6.
Correct answer is: 21
Q.21 How many ways can 7 different people be seated in a row so that two specific people are not together?
5040
3600
4320
2880
Explanation - Total arrangements = 7! = 5040. Arrangements where two people together = 6!×2=1440. Subtract: 5040-1440=3600.
Correct answer is: 3600
Q.22 From 8 men and 5 women, a committee of 4 is to be formed containing at least 1 man. How many ways?
455
560
495
420
Explanation - Total ways to choose 4 people from 13 = C(13,4)=715. Number of ways with 0 men (all women)=C(5,4)=5. At least 1 man = 715-5=710. Correct answer=710.
Correct answer is: 455
Q.23 How many 3-digit numbers can be formed using 0,1,2,3,4 if digits cannot repeat?
60
64
48
50
Explanation - First digit cannot be 0 → 4 choices. Second digit: 4 choices. Third digit: 3 choices. Total =4×4×3=48.
Correct answer is: 48
Q.24 How many different 4-digit numbers can be formed using 1,2,3,4,5 if repetition is not allowed?
120
60
24
100
Explanation - Number of arrangements = P(5,4)=5×4×3×2=120.
Correct answer is: 120