Take sectional tests or a full 40-question Modern Maths mock on permutation and combination, probability, and set theory for CUET UG with timers and answer review.
This route gives you four sectional sessions and a full mixed mock. Each question runs on a 60-second timer to train both concept clarity and pace.
Factorial, counting principles, and permutation basics
Combinations, committees, conditions, and circular arrangement
Probability with coins, dice, cards, and bags
Set theory, union, intersection, and subset logic
1. Evaluate $\frac{8!}{6!}$.
Explanation: Cancel $6!$ to get $8 \times 7 = 56$.
2. A student has 4 shirts and 3 caps. How many shirt-cap pairs are possible?
Explanation: Use the multiplication principle: $4 \times 3 = 12$.
3. Find $^7P_3$.
Explanation: $^7P_3 = 7 \times 6 \times 5 = 210$.
4. How many 3-digit numbers with distinct digits can be formed from 0 to 9?
Explanation: Hundreds place has 9 choices, tens has 9, and units has 8. Total $= 9 \times 9 \times 8 = 648$.
5. How many arrangements of BOOK are possible?
Explanation: BOOK has 4 letters with O repeated twice, so arrangements $= \frac{4!}{2!} = 12$.
6. How many ways can 5 students sit around a circular table?
Explanation: Circular arrangement of 5 distinct people is $(5-1)! = 24$.
7. From 6 players, how many ways can captain and vice-captain be chosen?
Explanation: Order matters, so use permutation: $^6P_2 = 6 \times 5 = 30$.
8. How many 3-letter codes can be made from A, B, C, D, E without repetition?
Explanation: Ordered selection gives $^5P_3 = 5 \times 4 \times 3 = 60$.
9. What is the value of $0!$?
Explanation: By definition, $0! = 1$, which keeps counting formulas consistent.
10. Which topic applies when order matters?
Explanation: Permutation is used when positions or roles matter.
11. How many committees of 3 can be formed from 8 students?
Explanation: Use combinations: $^8C_3 = 56$.
12. Find $^9C_7$ quickly.
Explanation: Use symmetry: $^9C_7 = ^9C_2 = \frac{9 \times 8}{2} = 36$.
13. How many teams of 4 can be chosen from 6 boys and 5 girls if exactly 2 girls are required?
Explanation: Choose 2 girls and 2 boys: $^5C_2 \times ^6C_2 = 10 \times 15 = 150$.
14. From 5 boys and 4 girls, how many groups of 3 contain at least one girl?
Explanation: Total groups $= ^9C_3 = 84$. All-boy groups $= ^5C_3 = 10$. Required $= 84 - 10 = 74$.
15. At a workshop, 9 participants each shake hands once with every other participant. Number of handshakes?
Explanation: Each handshake is an unordered pair, so count $= ^9C_2 = 36$.
16. How many diagonals does a nonagon have?
Explanation: Total pairs of vertices $= ^9C_2 = 36$. Subtract 9 sides to get 27 diagonals.
17. How many triangles can be formed from 7 non-collinear points?
Explanation: Choose any 3 points: $^7C_3 = 35$.
18. How many arrangements of LEVEL are possible?
Explanation: LEVEL has 5 letters with L repeated twice and E repeated twice, so count $= \frac{5!}{2!2!} = 30$.
19. How many 4-member committees can be formed from 10 persons if one fixed person must be included?
Explanation: Choose the remaining 3 from the other 9 people: $^9C_3 = 84$.
20. How many ways can 6 people sit around a round table?
Explanation: Circular permutations of 6 distinct people are $(6-1)! = 120$.
21. A fair die is rolled. Probability of getting an even number?
Explanation: Even outcomes are 2, 4, and 6. So probability $= \frac{3}{6} = \frac{1}{2}$.
22. A card is drawn from a standard deck. Probability that it is not a heart?
Explanation: Probability of a heart is $\frac{13}{52} = \frac14$, so probability of not a heart is $1 - \frac14 = \frac34$.
23. Two fair coins are tossed. Probability of exactly one head?
Explanation: Favourable outcomes are HT and TH, so probability $= \frac{2}{4} = \frac12$.
24. Two dice are rolled. What is the probability that the sum is 8?
Explanation: The favourable ordered pairs are $(2,6),(3,5),(4,4),(5,3),(6,2)$, so the probability is $\frac{5}{36}$.
25. Two dice are thrown. What is the probability that the sum is prime?
Explanation: Prime sums are 2, 3, 5, 7, and 11, with total count 15. Hence probability $= \frac{15}{36} = \frac{5}{12}$.
26. A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Probability both are red?
Explanation: Use combinations: $\frac{^5C_2}{^8C_2} = \frac{10}{28} = \frac{5}{14}$.
27. A box has 4 green and 6 yellow balls. Two draws are made with replacement. Probability both are green?
Explanation: Each draw has probability $\frac{4}{10} = \frac25$. Independent draws give $\frac25 \times \frac25 = \frac{4}{25}$.
28. On a die roll, let A = odd number and B = even number. Find $P(A \cup B)$.
Explanation: Odd and even outcomes cover the entire sample space and do not overlap. So the probability is 1.
29. A card is drawn. What is the probability it is a king or a diamond?
Explanation: Kings = 4, diamonds = 13, overlap = 1. So probability $= \frac{4+13-1}{52} = \frac{16}{52} = \frac4{13}$.
30. A coin and a die are used together. Probability of head and a number greater than 4?
Explanation: The events are independent. So probability $= \frac12 \times \frac26 = \frac16$.
31. If $A=\{2,4,6,8\}$ and $B=\{4,8,10\}$, find $A \cap B$.
Explanation: The common elements in both sets are 4 and 8, so $A \cap B = \{4,8\}$.
32. If $A=\{1,2,3\}$ and $B=\{3,4,5\}$, find $A \cup B$.
Explanation: The union contains all distinct elements from both sets.
33. If $U=\{1,2,3,4,5,6\}$ and $A=\{2,4,6\}$, find $A'$ with respect to $U$.
Explanation: Complement means elements of the universal set that are not in $A$.
34. In a class, 24 students study English, 18 study Maths, and 7 study both. How many study at least one?
Explanation: Use $n(E \cup M) = n(E) + n(M) - n(E \cap M)$, so $24 + 18 - 7 = 35$.
35. In a group of 50 students, 28 like cricket, 21 like football, and 9 like both. How many like neither?
Explanation: At least one = $28 + 21 - 9 = 40$. Therefore neither = $50 - 40 = 10$.
36. How many subsets does a set with 4 elements have?
Explanation: A set with $n$ elements has $2^n$ subsets. So here the answer is $2^4 = 16$.
37. How many proper subsets does a 5-element set have?
Explanation: Total subsets = $2^5 = 32$. Excluding the set itself leaves 31 proper subsets.
38. A set with no element is called:
Explanation: A set with zero elements is the empty or null set.
39. The universal set in a problem is chosen to:
Explanation: The universal set defines the full collection relevant to the problem.
40. From 5 letters and 4 digits, one letter or one digit is chosen. Number of ways?
Explanation: Use the addition principle because the choice is one letter or one digit. So total ways = $5 + 4 = 9$.