Counting Methods - ACT Math

1. How many ways can 5 different keys be arranged on a keyring?

   4. Circular arrangement with identical rotations.

2. What is the number of distinct permutations of the letters in $\text{LEVEL}$?

   $\frac{5!}{2!2!} = 30$. Two L's and two E's create identical arrangements.

3. What is the number of ways to choose $r$ items from $n$ distinct items when order does not matter?

   $^nC_r$. Selection without regard to arrangement order.

4. Identify the number of ways to arrange 4 people in a line.

   24. Calculate $4! = 4 \times 3 \times 2 \times 1 = 24$ arrangements.

5. Identify the rule: when choices are mutually exclusive, how do you count total outcomes?

   Add the counts: $m+n+\cdots$. Addition principle for disjoint events.

6. How many ways can 8 people sit in a circle?

   7. Fix one position to eliminate rotational symmetry.

7. How many ways can you choose 4 items from 10?

   $\binom{10}{4} = 210$. Combination formula with $n=10$ and $r=4$.

8. What is the formula for calculating combinations of $n$ items taken $r$ at a time?

   $C(n, r) = \frac{n!}{r!(n-r)!}$. Order doesn't matter, so divide by $r!$ arrangements.

9. How many ways can you select 3 desserts from a menu of 5?

   $\binom{5}{3} = 10$. Order doesn't matter for dessert selection.

10. What is the number of distinct permutations of $n$ objects with repeats $n_1,n_2,\dots,n_k$?

    $\frac{n!}{n_1!n_2!\cdots n_k!}$. Divide by factorials of identical object counts.

11. How many ways can you distribute 7 different books to 7 people?

    7. Each book goes to exactly one person.

12. What is $^6C_3$?

    $20$. $\frac{6!}{3!3!} = \frac{720}{36} = 20$

13. What is the principle of multiplication in counting?

    If one event has $m$ outcomes and another has $n$, total is $m \times n$. Sequential events multiply their outcome counts.

14. What is the formula for permutations with repetition?

    $n^r$. Each position can be filled with any of $n$ items.

15. What is the number of binary strings of length $5$ with exactly $2$ ones?

    $^5C_2 = 10$. Choose positions for ones among five spots.

16. How many different 4-digit numbers can be formed using the digits 1-9?

    $9 \times 9 \times 8 \times 7 = 4,536$. First digit excludes 0, others exclude previous digits.

17. Identify the number of ways to choose 0 objects from a set of 7.

    1. By definition, $C(n,0) = 1$ for any non-negative integer $n$.

18. What is the number of permutations where 3 items are identical and 2 are distinct?

    $\frac{5!}{3!}$. Divides by $3!$ to account for identical items in permutations.

19. What is the formula for combinations of $r$ objects from $n$ objects?

    $C(n, r) = \frac{n!}{r!(n-r)!}$. Divides by $r!$ to eliminate order since combinations ignore arrangement.

20. What is the formula for circular permutations of $n$ distinct objects?

    $(n-1)!$. Fix one object to break circular symmetry.

21. What is the number of 3-letter strings from $26$ letters if repetition is allowed?

    $26^3$. Each position chosen independently with replacement.

22. What is the Fundamental Counting Principle for $m$ choices followed by $n$ choices?

    $m \cdot n$ total outcomes. Multiply choices at each step for sequential decisions.

23. What is the number of permutations of $n$ distinct objects taken all at once?

    $n!$. Each position has decreasing choices available.

24. What is the value of $1!$?

    $1! = 1$. Only one way to arrange a single object.

25. How many ways can 4 different books be arranged on a shelf if 2 are always together?

    12. Treat the pair as one unit: $3! \times 2! = 12$ arrangements.

26. What is the number of ways to assign president and vice president from 6 people?

    $^6P_2 = 30$. Officer positions require specific ordering.

27. What is $^5P_5$?

    $120$. All arrangements of $5$ distinct objects: $5! = 120$

28. What is the number of ways to arrange 8 objects in a circle?

    5040. Circular arrangements use $(n-1)! = 7! = 5040$ for 8 objects.

29. How many ways can 2 identical balls be placed in 3 different boxes?

    6. Stars and bars: $C(2+3-1,3-1) = C(4,2) = 6$ placements.

30. What is the number of 3-letter strings from $26$ letters if repetition is not allowed?

    $^ {26}P_3$. Permutation formula for distinct selections.

31. What is the two-set inclusion–exclusion formula for $|A \cup B|$?

    $|A \cup B| = |A| + |B| - |A \cap B|$. Add sets, subtract overlap to avoid double-counting.

32. What does $n!$ represent in counting methods?

    The product of all positive integers up to $n$. Calculated as $n \times (n-1) \times ... \times 2 \times 1$.

33. How many ways can a president and vice president be selected from 5 candidates?

    20. Order matters for president and vice president roles.

34. What is a combination?

    A selection of items without regard to order. Order is not important in selections.

35. How many ways can you arrange the letters in the word 'BOOK'?

    $\frac{4!}{2!} = 12$. Two O's are identical, so divide by $2!$.

36. How many distinct permutations can be made from the letters in 'LEVEL'?

    30. Calculate $\frac{5!}{2!2!} = \frac{120}{4} = 30$ with repeated letters.

37. In a group of 10 people, how many ways can you choose a committee of 3?

    $\binom{10}{3} = 120$. Order doesn't matter when selecting committee members.

38. State the number of ways to arrange 3 books on a shelf.

    6. Three distinct books can be arranged in $3! = 6$ ways.

39. How many ways can 3 objects be selected from 5 distinct objects?

    10. Using $C(5,3) = \frac{5!}{3!2!} = \frac{120}{6 \cdot 2} = 10$.

40. What is the formula for calculating permutations of $n$ items taken $r$ at a time?

    $P(n, r) = \frac{n!}{(n-r)!}$. Order matters, so divide by unused positions $(n-r)!$.

41. What is the factorial definition of $n!$ for a positive integer $n$?

    $n! = n(n-1)(n-2)\cdots 2\cdot 1$. Product of all positive integers from $n$ down to $1$.

42. What is the formula for the number of subsets of a set with $n$ elements?

    $2^n$. Each element can either be included or excluded.

43. What is the complement rule for counting: how do you count outcomes in set $A$ using $A^c$?

    $|A| = |U| - |A^c|$. Count complement, subtract from universal set.

44. How many ways can you form a group of 4 from 10 people?

    $\binom{10}{4} = 210$. Same as choosing 4 from 10 people.

45. What is $^5C_5$?

    $1$. One way to choose everything from any set.

46. If a lock has 3 dials with 10 digits each, how many combinations are possible?

    $10^3 = 1,000$. Each dial has 10 choices, multiply together.

47. What is $ ^8C_2 $?

    $28$. $\frac{8!}{2!6!} = \frac{8 \cdot 7}{2} = 28$

48. What is the value of $5!$?

    120. Compute $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

49. What is $ ^7P_3 $?

    $210$. $\frac{7!}{4!} = \frac{5040}{24} = 210$

50. How many different outcomes are possible when tossing 3 coins?

    $2^3 = 8$. Each coin has 2 outcomes, multiply together.

51. What is $ {}^7 $P_3$ $?

    $210$. $\frac{7!}{4!} = \frac{5040}{24} = 210$

52. If a password consists of 4 digits, how many possible passwords are there?

    $10^4 = 10,000$. Each digit has 10 choices, multiply for all positions.

53. How many ways can 4 identical items be distributed into 3 different boxes?

    15. Stars and bars: $C(4+3-1,3-1) = C(6,2) = 15$ distributions.

54. What is a permutation?

    An arrangement of items in a specific order. Order is important in arrangements.

55. What is the number of ways to arrange $n$ distinct objects in a circle (rotations considered identical)?

    $(n-1)!$. Fix one position to eliminate rotational symmetry.

56. How many different 3-person committees can be formed from 7 people?

    35. Calculate $C(7,3) = \frac{7!}{3!4!} = \frac{210}{6} = 35$ committees.

57. What is the number of ways to choose a 2-person committee from 6 people?

    $ {}^6C_2 = 15 $. Committee selection ignores member ordering.

58. What is the number of outcomes when rolling a die and flipping a coin once?

    $6\cdot 2 = 12$. Independent events multiply their outcome counts.

59. If there are 3 routes from A to B and 2 from B to C, how many routes from A to C?

    6. Multiply choices for each segment of the journey.

60. What is $ ^6C_3 $?

    $20$. $\frac{6!}{3!3!} = \frac{720}{36} = 20$

61. How many ways can 3 identical balls be placed in 4 different boxes?

    20. Stars and bars: $C(3+4-1,4-1) = C(6,3) = 20$ placements.

62. How many permutations are possible for arranging $n$ distinct objects?

    $n!$. Each position has decreasing choices, resulting in $n!$ total arrangements.

63. What is the number of ways to distribute 5 identical apples to 3 children?

    21. Stars and bars: $C(5+3-1,3-1) = C(7,2) = 21$ ways.

64. What is the number of subsets of a set with $n$ elements?

    $2^n$. Each element can be included or excluded.

65. If you have 4 shirts and 3 pants, how many outfits can you create?

    12. Multiply clothing options using multiplication principle.

66. What is $|A \cup B|$ if $|A|=12$, $|B|=9$, and $|A\cap B|=5$?

    $16$. Using inclusion-exclusion: $12 + 9 - 5 = 16$

67. How many ways can the letters in 'ACT' be arranged?

    6. Three distinct letters give $3! = 6$ different arrangements.

68. What is the formula for the number of ways to arrange $n$ objects where some are alike?

    $\frac{n!}{p_1!p_2!...p_k!}$. Divide by factorials of repeated object counts.

69. What is the formula for combinations of $n$ distinct objects chosen $r$ at a time?

    $^{n}C_r = \frac{n!}{r!(n-r)!}$. Divides out arrangements within selected group.

70. What is $^5C_0$?

    $1$. One way to choose nothing from any set.

71. What is the factorial of 5?

    120. $5! = 5 \times 4 \times 3 \times 2 \times 1$.

72. How many ways can a committee of 2 be formed from 5 people?

    $\binom{5}{2} = 10$. Order doesn't matter for committee selection.

73. What is the number of binary strings of length $n$ (each position $0$ or $1$)?

    $2^n$. Each position has $2$ independent choices.

74. What is the binomial coefficient notation for combinations?

    $\binom{n}{r}$. 'n choose r' notation for combination calculations.

75. What is the number of circular arrangements of $7$ distinct people around a round table?

    $(7-1)! = 720$. Fix one person's position to eliminate rotations.

76. What is the value of $0!$?

    1. By definition, factorial of zero equals one.

77. What is the formula to calculate the number of subsets of a set?

    $2^n$. Each element can be included or excluded from any subset.

78. What is the factorial of a non-negative integer $n$ represented by?

    $n!$. The symbol for the product of all positive integers from 1 to $n$.

79. State the number of ways to assign 3 tasks to 3 workers, one task per worker.

    6. One-to-one assignment gives $3! = 6$ possible arrangements.

80. What is the number of combinations of 10 objects taken 4 at a time?

    210. Using $C(10,4) = \frac{10!}{4!6!} = \frac{5040}{24} = 210$ combinations.

81. How many ways can you choose 2 cards from a deck of 52?

    $\binom{52}{2} = 1,326$. Order doesn't matter when choosing cards.

82. How many distinct permutations are there of the letters in 'MISSISSIPPI'?

    $\frac{11!}{4!4!2!} = 34,650$. Divide by repeated letters: 4 I's, 4 S's, 2 P's.

83. What is the number of distinct permutations of the letters in $\text{MISS}$?

    $\frac{4!}{2!} = 12$. Two S's create identical arrangements.

84. How many ways can you arrange 3 red balls and 2 blue balls in a row?

    $\frac{5!}{3!2!} = 10$. Divide by repetitions of identical colored balls.

85. What is the formula for permutations of $n$ distinct objects taken $r$ at a time?

    $^nP_r = \frac{n!}{(n-r)!}$. Accounts for unused objects in denominator.

86. State the number of ways to choose 2 items from a set of 6.

    15. Using $C(6,2) = \frac{6!}{2!4!} = \frac{30}{2} = 15$ ways.

87. What is the number of subsets of a set with $6$ elements?

    $2^6 = 64$. Each element independently included or excluded.

88. What is the number of ways to choose $r$ items from $n$ distinct items when order matters?

    $^nP_r$. Selection where arrangement order matters.

89. What is the number of outcomes for $k$ independent steps with $n_1,n_2,\dots,n_k$ choices each?

    $n_1 n_2 \cdots n_k$. Multiply choices at each independent step.

90. What is $^9P_2$?

    $72$. $\frac{9!}{7!} = 9 \cdot 8 = 72$

91. How many ways to arrange 6 people in a line if 2 must stand together?

    5! $\times 2$. Treat the pair as one unit, then arrange.

92. How many ways can a president and a vice-president be selected from 5 people?

    20. Use permutations: $P(5,2) = \frac{5!}{3!} = 20$ ordered selections.

93. Which formula calculates permutations of $r$ objects from $n$ objects?

    $P(n, r) = \frac{n!}{(n-r)!}$. Removes $(n-r)!$ to account for unfilled positions in permutations.

94. How many different 3-letter codes can be formed using the letters A, B, C, D?

    $4^3 = 64$. Each position has 4 letter choices with repetition.

95. Identify the number of ways to arrange 6 people if 2 must sit together.

    240. Treat the pair as one unit: $5! \times 2! = 240$ arrangements.

96. How many ways to arrange 3 red, 2 blue, and 1 green ball in a row?

    $\frac{6!}{3!2!1!} = 60$. Divide by repetitions of each color group.

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

    24. Number of ways to arrange 4 distinct objects.

98. What is the number of permutations of 5 objects taken 2 at a time?

    20. Apply $P(5,2) = \frac{5!}{3!} = \frac{120}{6} = 20$ permutations.

99. What is the relationship between permutations and combinations: $ ^nP_r $ in terms of $ ^nC_r $?

    $^nP_r = ^nC_r \cdot r!$. Permutations equal combinations times internal arrangements.

100. How many subsets can be formed from a set with 4 elements?

     16. Each element can be in or out, giving $2^4 = 16$ subsets.

101. What is $^8C_2$?

     $28$. $\frac{8!}{2!6!} = \frac{8 \cdot 7}{2} = 28$

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