1. How many ways can 3 students be selected and arranged from 5 students, if order matters and no repetition?

A. \( P(5,3) = \frac{5!}{(5-3)!} = 60 \) B. \( C(5,3) = \frac{5!}{3!2!} = 10 \) C. \( 5^3 = 125 \) D. \( 3^5 = 243 \)

2. In how many ways can 4-digit PIN codes be formed using digits 0–9 if digits may be repeated?

A. \(10^4 = 10,000\) B. \(10!\) C. \(P(10,4)\) D. \(C(10,4)\)

3. What principle states that if one event can occur in \( m \) ways and a second independent event can occur in \( n \) ways, then both events can occur in \( m \times n \) ways?

A. Addition principle B. Fundamental counting principle (rule of product) C. Permutation principle D. Combination principle

4. How many ways can 3 fruits be selected from 5 types if order matters and repetition is NOT allowed?

A. \( P(5,3) = \frac{5!}{2!} = 60 \) B. \( C(5,3) = 10 \) C. \( 5^3 = 125 \) D. \( C(7,3) = 35 \)

5. If order matters and repetition is allowed, how many permutations can be formed by choosing r objects from n distinct objects?

A. \( \frac{n!}{(n-r)!} \) B. \( n^r \) C. \( \frac{n!}{r!(n-r)!} \) D. \( n! \)

6. In a permutation problem without repetition, what happens as the number of positions r approaches the total number of objects n?

A. Number of permutations approaches 1 B. Number of permutations equals \(n!\) C. Number of permutations equals \(n^r\) D. Number of permutations equals \(C(n,r)\)

7. In a combination with repetition allowed, the formula to find the number of combinations is:

A. \( C(n,r) = \frac{n!}{r!(n-r)!} \) B. \( P(n,r) = \frac{n!}{(n-r)!} \) C. \( C(n+r-1, r) = \frac{(n+r-1)!}{r!(n-1)!} \) D. \( n^r \)

8. Which of the following best describes when to use combinations?

A. When order matters and repetition is allowed B. When order matters and repetition is not allowed C. When order does not matter and repetition is not allowed D. When order does not matter and repetition is always allowed

9. How many 3-letter passwords can be formed from letters A, B, C, D if letters may be repeated and order matters?

A. 64 B. 24 C. 12 D. 4

10. What is the key condition for applying combinations with repetition?

A. Order matters, no repetition B. Order matters, repetition allowed C. Order does not matter, repetition allowed D. Order does not matter, no repetition

11. How is a combination defined in counting?

A. Arrangement where order matters B. Arrangement where order does not matter C. Arrangement that allows repetition only D. Arrangement with fixed order but no repetition

12. Which of the following is true regarding permutations?

A. Order does not matter B. Repetition is always allowed C. Order matters D. It can be used only for unlimited objects

13. What is the formula for the number of permutations with repetition allowed for \( n \) objects arranged in \( r \) positions?

A. \( P(n,r) = \frac{n!}{(n-r)!} \) B. \( n^r \) C. \( C(n,r) = \frac{n!}{r!(n-r)!} \) D. \( C(n+r-1, r) \)

14. How many different 3-letter passwords can be formed from letters A, B, C, D if repetition is allowed?

A. \(4^3 = 64\) B. \(4! = 24\) C. \(\frac{4!}{(4-3)!} = 24\) D. \(\frac{4!}{3!} = 4\)

15. Which formula calculates the number of permutations of \( r \) objects selected from \( n \) distinct objects without repetition?

A. \( P(n,r) = \frac{n!}{(n-r)!} \) B. \( P(n,r) = \frac{n!}{r!(n-r)!} \) C. \( C(n,r) = \frac{n!}{r!(n-r)!} \) D. \( C(n,r) = \frac{(n+r-1)!}{r!(n-1)!} \)

16. If 4 students are seated in 4 distinct chairs, how many distinct arrangements are there?

A. 4 B. 16 C. 24 D. 256

17. Which of the following best describes a permutation problem?

A. Selecting a group where order does not matter B. Counting the number of ways to arrange objects where order matters C. Selecting objects with possible repetition without ordering D. Counting only subsets of objects without arranging

18. How many ways can 4 books be arranged on a shelf?

A. 16 B. 24 C. 12 D. 256

19. Why is it important to distinguish whether repetition is allowed in counting problems?

A. It affects whether order matters or not B. Different formulas apply for repetition and no repetition C. Repetition only affects combinations D. It does not change the total number of possibilities

20. Which statement is true about combinations with repetition allowed?

A. The order of selection matters B. Repetition of elements is not allowed C. The formula used is \( C(n+r-1, r) \) D. It uses the formula \( \frac{n!}{(n-r)!} \)

21. If a team of 3 students is selected from 8 students and the order does not matter, which formula should be used?

A. \( 8^3 \) B. \( P(8,3) \) C. \( C(8,3) \) D. \( C(8+3-1, 3) \)

22. When repetition is allowed but order matters, which counting method is used?

A. Permutations without repetition B. Combinations without repetition C. Permutations with repetition D. Combinations with repetition

23. When choosing r objects from n with order not mattering and no repetition, the number of ways is denoted by:

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

24. What is the formula for combinations without repetition?

A. \( C(n,r) = \frac{n!}{r!(n-r)!} \) B. \( P(n,r) = \frac{n!}{(n-r)!} \) C. \( C(n,r) = \frac{(n+r-1)!}{r!(n-1)!} \) D. \( P(n,r) = n^r \)

25. Selecting 3 students from 5 to form a committee (order does not matter, no repetition) has how many possible selections?

A. 60 B. 10 C. 120 D. 125

26. Which of the following is true about permutations without repetition?

A. Order does not matter and repetition is allowed B. Order matters and repetition is not allowed C. Order does not matter and repetition is not allowed D. Order matters and repetition is allowed

27. In counting combinations where order does not matter and repetition is allowed, why is \( C(n+r-1, r) \) used?

A. Because selections are sequences B. Because it counts combinations allowing identical elements multiple times C. Because the order of selection is changed D. Because repetition is disallowed

28. For determining the number of different 4-digit PIN codes from digits 0–9 with repetition allowed, which formula is used?

A. \( 10! \) B. \( 10^4 = 10000 \) C. \( P(10,4) = \frac{10!}{6!} = 5040 \) D. \( C(10,4) \)

29. Select the correct statement:

A. \(P(n,r)\) counts the number of ways to select r objects from n when order does not matter B. \(C(n,r)\) counts the number of ways to arrange r objects from n when order matters C. \(P(n,r) = \frac{n!}{(n-r)!}\) counts arrangements where order matters and no repetition D. \(C(n,r) = \frac{n!}{(n-r)!}\) counts combinations with repetition

30. What does the formula \( C(n+r-1, r) \) represent in counting?

A. Permutations without repetition B. Permutations with repetition C. Combinations without repetition D. Combinations with repetition

31. How many ways can 3 sweets be selected from 5 types if repetition is allowed and order does not matter?

A. 60 B. 10 C. 35 D. 125

32. If order matters and repetition is allowed, how many 3-letter arrangements are possible using letters A, B, C, D?

A. 64 B. 12 C. 24 D. 10

33. What is a common mistake when solving counting problems?

A. Forgetting to multiply independent events B. Always assuming order matters C. Using permutations when order actually does not matter D. Using combinations when repetition is allowed

34. When selecting a committee where order does not matter and repetition is not allowed, which formula should be used to find the number of ways to select r people from n?

A. \(P(n,r) = \frac{n!}{(n-r)!}\) B. \(n^r\) C. \(C(n,r) = \frac{n!}{r!(n-r)!}\) D. \(C(n+r-1, r)\)

35. What is the fundamental counting principle (rule of product) used to calculate?

A. The number of permutations when repetition is not allowed B. The total number of possible outcomes when combining independent events C. The number of combinations without repetition D. The number of permutations with repetition

36. How does repetition affect the formula for combinations?

A. It changes \( C(n,r) \) to \( C(n+r-1, r) \) B. It changes \( P(n,r) \) to \( n^r \) C. It does not affect the formula D. It changes \( P(n,r) \) to \( \frac{n!}{(n-r)!} \)

37. How many different 3-student teams can be formed from 5 students if order matters and repetition is not allowed?

A. \( P(5,3) = \frac{5!}{2!} = 60 \) B. \( C(5,3) = 10 \) C. \( 5^3 = 125 \) D. \( C(5+3-1, 3) = 35 \)

38. For a selection of a team of 3 students from 8 students, where order does not matter and no repetition is allowed, which formula should be used?

A. \( P(8,3) = \frac{8!}{5!} = 336 \) B. \( C(8,3) = \frac{8!}{3!5!} = 56 \) C. \( 8^3 = 512 \) D. \( C(10,3) \)

39. If order matters and repetition is not allowed, how many ways are there to arrange r objects from n?

A. \( n^r \) B. \( \frac{n!}{(n-r)!} \) C. \( C(n,r) \) D. \( \frac{n!}{r!} \)

40. Why should one clearly identify if order matters and if repetition is allowed before applying formulas in counting problems?

A. To use factorials correctly B. To select the most complicated formula C. Because these factors determine the correct counting method and formula D. To simplify complex calculations only

41. How many ways can 4 students be seated in 4 distinct chairs if each student must occupy exactly one chair?

A. \( 4! = 24 \) B. \( 4^4 = 256 \) C. \( P(4,2) = 12 \) D. \( C(4,4) = 1 \)

42. Which of the following is NOT a common mistake in permutation and combination problems?

A. Using permutations where order does not matter B. Forgetting whether repetition is allowed C. Correctly identifying the formula \( P(n,r) \) vs. \( C(n,r) \) D. Confusing permutations and combinations

43. What does a permutation represent in counting problems?

A. Selection without order B. Selection where order matters C. Selection where repetition is not allowed D. Selection where repetition is always allowed

44. What is the incorrect way of describing a combination with repetition?

A. Order does not matter B. Objects can be chosen more than once C. Order matters D. Uses the formula \( C(n+r-1,r) \)

45. When all \( n \) objects are arranged, the total number of permutations is:

A. \( n^n \) B. \( n! \) C. \( \frac{n!}{2!} \) D. \( \frac{n!}{(n-r)!} \)

46. Why must one avoid using permutations when order does not matter?

A. Permutations give fewer possibilities than combinations B. Permutations ignore repetition C. Permutations count arrangements where order matters, which inflates the number of outcomes D. Permutations are only for identical objects

47. What is the primary difference between permutations and combinations?

A. Repetition allowed or not B. Whether order matters or not C. Number of objects available D. Problem complexity

48. In a problem where the same type of fruit can be selected multiple times, and the order of selection does not matter, which counting method applies?

A. Combination without repetition B. Permutation with repetition C. Combination with repetition D. Permutation without repetition

49. The number of ways to arrange n distinct objects in all possible orders is:

A. \( n^n \) B. \( n! \) C. \( \frac{n!}{(n-r)!} \) D. \( \frac{n!}{r!(n-r)!} \)

50. If a cafeteria has 5 fruit types and a student selects 3 fruits with repetition allowed and order does not matter, how many combinations exist?

A. \( C(5,3) = 10 \) B. \( P(5,3) = 60 \) C. \( C(5+3-1, 3) = C(7,3) = 35 \) D. \( 5^3 = 125 \)