1. What is the value of \(5!\)?

A. 60 B. 120 C. 24 D. 720

2. Which of the following expresses De Morgan's law for the complement of union of two sets \(A\) and \(B\)?

A. \(A \cup B\) B. \(A \cap B\) C. \(\overline{A} \cap \overline{B}\) D. \(\overline{A} \cup \overline{B}\)

3. Given equally likely outcomes, how do you calculate the probability \(P(E)\) of an event \(E\) from sample space \(S\)?

A. \(|E| \times |S|\) B. \(|E| / |S|\) C. \(|S| / |E|\) D. \(|E| - |S|\)

4. What does strong induction assume?

A. \(P(k)\) is true B. \(P(1), P(2), ..., P(k)\) are true C. \(P(k+1)\) is true D. \(P(n)\) is true for all \(n\)

5. Two events \(E\) and \(F\) are independent if:

A. \(P(E \cap F) = P(E) + P(F)\) B. \(P(E \cap F) = P(E) \times P(F)\) C. \(P(E|F) = 0\) D. \(P(E \cup F) = P(E) + P(F)\)

6. What is the expansion of \(\sum_{r=0}^n C(n,r)\)?

A. \(n^2\) B. \(2^n\) C. \(n!\) D. \(2n\)

7. What is the sum of all the combinations \(C(n, r)\) for \(r\) ranging from 0 to \(n\)?

A. \(n^2\) B. \(2^n\) C. \(n!\) D. \(2n\)

8. How many circular permutations are there for \(n\) distinct objects?

A. \(n!\) B. \((n-1)!\) C. \(\frac{n!}{2}\) D. \((n+1)!\)

9. A function is injective if and only if:

A. \(f(a_1) = f(a_2) \Rightarrow a_1 = a_2\) B. \(f(a_1) = f(a_2) \Rightarrow a_1 \neq a_2\) C. \(a_1 = a_2 \Rightarrow f(a_1) \neq f(a_2)\) D. Every element in the codomain is mapped

10. The sum of the first 10 odd numbers equals:

A. 100 B. 90 C. 110 D. 120

11. How many 3-member committees can be formed from 8 people?

A. 336 B. 56 C. 24 D. 512

12. The generalized pigeonhole principle ensures that among 13 people, at least how many share a birth month?

A. 1 B. 2 C. 3 D. 4

13. How do you calculate conditional probability \(P(E|F)\)?

A. \(\frac{P(E \cup F)}{P(F)}\) B. \(\frac{P(E) \times P(F)}{P(F)}\) C. \(\frac{P(E \cap F)}{P(F)}\) D. \(\frac{P(E)}{P(F)}\)

14. The sum of the first \(n\) odd numbers is:

A. \(n^2\) B. \(n(n+1)\) C. \(2n\) D. \(n\)

15. Given the recurrence relation aₙ = 3a_{n-1} - 2a_{n-2}, what is the characteristic equation associated with this relation?

A. r² = 3r - 2 B. r² = 3r + 2 C. r² - 3r + 2 = 0 D. r² + 3r - 2 = 0

16. How many combinations are there when selecting 3 members from 8 people?

A. 336 B. 56 C. 24 D. 512

17. Which of the following is a tautology?

A. \(p \wedge \neg p\) B. \(p \to (p \vee q)\) C. \(p \wedge q\) D. \(p \oplus q\) (exclusive or)

18. What is the probability of getting a sum of 7 when rolling two dice?

A. \(\frac{1}{6}\) B. \(\frac{1}{12}\) C. \(\frac{5}{36}\) D. \(\frac{1}{9}\)

19. In set theory, \(|\mathcal{P}(A)| = ?\) where \(|A|=n\)

A. \(n\) B. \(2n\) C. \(2^n\) D. \(n^2\)

20. What is the expanded form of \((1-x)^4\)?

A. \(1-4x + 6x^2 - 4x^3 + x^4\) B. \(1 + 4x + 6x^2 + 4x^3 + x^4\) C. \(1 - 4x - 6x^2 - 4x^3 - x^4\) D. \(1 + 4x - 6x^2 + 4x^3 - x^4\)

21. The factorial 5! equals:

A. 60 B. 120 C. 24 D. 720

22. How many rows are there in a truth table for 3 variables?

A. 4 B. 6 C. 8 D. 16

23. When is a biconditional statement \(p \leftrightarrow q\) true?

A. When \(p\) and \(q\) have different truth values B. When \(p\) is true and \(q\) is false C. When \(p\) and \(q\) have the same truth value D. When \(p\) is false and \(q\) is true

24. Which of the following expansions correctly represents \((1 - x)^4\)?

A. \(1 - 4x + 6x^2 - 4x^3 + x^4\) B. \(1 + 4x + 6x^2 + 4x^3 + x^4\) C. \(1 - 4x - 6x^2 - 4x^3 - x^4\) D. \(1 + 4x - 6x^2 + 4x^3 - x^4\)

25. What is the coefficient of \(x^3\) in the expansion of \((1+x)^5\)?

A. 5 B. 10 C. 15 D. 20

26. For the binomial expansion of \((x+y)^6\), what is the coefficient of the middle term?

A. 15 B. 20 C. 10 D. 30

27. When is the implication \(p \to q\) false?

A. When both \(p\) and \(q\) are true B. When \(p\) is true and \(q\) is false C. When \(p\) is false and \(q\) is true D. When both \(p\) and \(q\) are false

28. The pigeonhole principle implies that if 10 pigeons are placed in 9 boxes, then:

A. Each box has exactly one pigeon B. At least one box has at least two pigeons C. All boxes are empty D. Some box has zero pigeons

29. What method is used to prove the formula \(1 + 2 + \dots + n = \frac{n(n+1)}{2}\)?

A. Direct proof B. Contradiction C. Induction D. Cases

30. How many distinct circular permutations can be made from \(n\) objects?

A. \(n!\) B. \((n-1)!\) C. \(n!/2\) D. \((n+1)!\)

31. What is an injective function?

A. Every element in codomain is mapped B. \(f(a_1) = f(a_2) \implies a_1 = a_2\) C. \(f(a_1) = f(a_2) \implies a_1 \neq a_2\) D. None of these

32. What is the correct formula for the probability \(P(E)\) for equally likely outcomes?

A. \(|E| \times |S|\) B. \(\frac{|E|}{|S|}\) C. \(\frac{|S|}{|E|}\) D. \(|E| - |S|\)

33. What is the negation of the statement "All students passed"?

A. No students passed B. Some students passed C. Some students did not pass D. All students failed

34. The number of ways to arrange the letters of "MISSISSIPPI" is:

A. \(11!\) B. \(\frac{11!}{4!4!2!}\) C. \(\frac{11!}{4!4!}\) D. \(4!4!2!\)

35. What is the number of permutations \(P(10, 3)\) (arrangements of 3 out of 10 objects)?

A. 720 B. 120 C. 30 D. 1000

36. What is the principle behind the pigeonhole principle?

A. If \(n\) items are placed into \(m\) boxes, then some box has at least \(\left\lceil \frac{n}{m} \right\rceil\) items B. Each box will have exactly \(n/m\) items C. Items are distributed equally in all boxes D. None of the above

37. Which principle states that if \(n\) items are placed into \(m\) boxes, then some box has at least:

A. \(\lfloor n/m \rfloor\) items B. \(\lceil n/m \rceil\) items C. \(n - m\) items D. \(n/m\) items

38. What defines a proposition in logic?

A. A command or request B. A question C. A declarative sentence with a definite truth value D. A mathematical expression

39. What is Pascal's Identity?

A. \(C(n,r) = C(n-1,r) + C(n-1,r-1)\) B. \(C(n,r) = C(n,r-1) + C(n-1,r)\) C. \(C(n,r) = C(n-1,r) + C(n,r-1)\) D. \(C(n,r) = C(n+1,r) - C(n,r)\)

40. How many arrangements are possible for the word “MISSISSIPPI”?

A. \(11!\) B. \(\frac{11!}{4!4!2!}\) C. \(\frac{11!}{4!4!}\) D. \(4!4!2!\)

41. The sum of a geometric series \(\sum_{i=0}^n r^i\) where \(r \neq 1\) equals:

A. \(\frac{r^{n+1} - 1}{r - 1}\) B. \(\frac{r^n - 1}{r - 1}\) C. \(\frac{1 - r^{n+1}}{1 - r}\) D. Both A and C

42. Pascal's identity for combinations \(C(n, r)\) can be expressed as:

A. \(C(n-1, r) + C(n-1, r-1)\) B. \(C(n, r-1) + C(n-1, r)\) C. \(C(n-1, r) + C(n, r-1)\) D. \(C(n+1, r) - C(n, r)\)

43. What type of mathematical induction is used to prove the statement "Every integer \(\geq 2\) is a product of primes"?

A. Regular induction B. Strong induction C. Contradiction D. Direct proof

44. What is the formula for combinations with repetition?

A. \(C(n, r)\) B. \(C(n + r - 1, r)\) C. \(P(n, r)\) D. \(n^r\)

45. The binomial theorem for \((a+b)^n\) is expressed as:

A. \(\sum C(n, r) a^r b^{n-r}\) B. \(\sum C(n, r) a^{n-r} b^r\) C. \(\sum P(n, r) a^{n-r} b^r\) D. \(\sum n! a^{n-r} b^r\)

46. What is the sum of the first \(n\) integers, \(1 + 2 + \dots + n\)?

A. \(\frac{n(n-1)}{2}\) B. \(n^2\) C. \(\frac{n(n+1)}{2}\) D. \(n(n+1)\)

47. What is the coefficient of the middle term in the expansion of \((x + y)^6\)?

A. 15 B. 20 C. 10 D. 30

48. Which of the following correctly describes the property of expectation for two random variables X and Y?

A. E(X + Y) = E(X) × E(Y) B. E(X + Y) = E(X) / E(Y) C. E(X + Y) = E(X) + E(Y) D. E(X + Y) = E(X) - E(Y)

49. Which of the following is a sufficient condition for the independence of two events \(E\) and \(F\)?

A. \(P(E|F) = P(E)\) B. \(P(E \cup F) = P(E) + P(F)\) C. \(P(E \cap F) = 0\) D. \(P(E|F) = 0\)

50. What is the fourth row (starting from row 0) in Pascal's triangle?

A. 1 3 3 1 B. 1 4 6 4 1 C. 1 5 10 10 5 1 D. 1 2 1