ACT Math Flashcards: Counting Methods

What this deck covers

This deck focuses on Counting Methods, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

Answer:

1. Circular arrangement with identical rotations.

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

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

Flashcard 3: How many permutations are possible for arranging $n$ distinct objects?

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

Flashcard 4: State the number of ways to assign 3 tasks to 3 workers, one task per worker.

Answer:

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

Flashcard 5: What is the number of binary strings of length $5$ with exactly $2$ ones?

Answer: $\binom{5}{2} = 10$. Choose positions for ones among five spots.

Flashcard 6: What is the number of ways to choose $r$ items from $n$ distinct items when order does not matter?

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

Flashcard 7: What is the number of permutations where 3 items are identical and 2 are distinct?

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

Flashcard 8: State the number of ways to arrange 3 books on a shelf.

Answer:

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

Flashcard 9: How many ways can you distribute 7 different books to 7 people?

Answer:

1. Each book goes to exactly one person.

Flashcard 10: What is the formula for circular permutations of $n$ distinct objects?

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

Flashcard 11: What does $n!$ represent in counting methods?

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

Flashcard 12: Identify the number of ways to arrange 4 people in a line.

Answer:

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

Flashcard 13: What is the number of ways to choose a 2-person committee from 6 people?

Answer: $^6C_2 = 15$. Committee selection ignores member ordering.

Flashcard 14: How many distinct permutations are there of the letters in 'MISSISSIPPI'?

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

Flashcard 15: What is the number of subsets of a set with $6$ elements?

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

Flashcard 16: Identify the rule: when choices are mutually exclusive, how do you count total outcomes?

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

Flashcard 17: How many ways can 8 people sit in a circle?

Answer:

1. Fix one position to eliminate rotational symmetry.

Flashcard 18: What is the two-set inclusion–exclusion formula for $|A \cup B|$?

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

Flashcard 19: What is the value of $0!$?

Answer:

1. By definition, factorial of zero equals one.

Flashcard 20: How many ways can you choose 4 items from 10?

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

Flashcard 21: What is the formula for calculating combinations of $n$ items taken $r$ at a time?

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

Flashcard 22: How many ways can you select 3 desserts from a menu of 5?

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

Flashcard 23: How many ways can you arrange 3 red balls and 2 blue balls in a row?

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

Flashcard 24: Identify the number of ways to arrange 4 people in a line.

Answer:

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

Flashcard 25: How many ways can a committee of 2 be formed from 5 people?

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

Flashcard 26: What is the number of distinct permutations of $n$ objects with repeats $n_1,n_2,\dots,n_k$?

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

Flashcard 27: What is $|A \cup B|$ if $|A|=12$, $|B|=9$, and $|A\cap B|=5$?

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

Flashcard 28: How many ways can 4 identical items be distributed into 3 different boxes?

Answer:

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

Flashcard 29: How many ways can you distribute 7 different books to 7 people?

Answer:

1. Each book goes to exactly one person.

Flashcard 30: What is $^6C_3$?

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

Flashcard 31: What is the relationship between permutations and combinations: $^nP_r$ in terms of $^nC_r$?

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

Flashcard 32: What is a permutation?

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

Flashcard 33: What is the formula for calculating permutations of $n$ items taken $r$ at a time?

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

Flashcard 34: What is the principle of multiplication in counting?

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

Flashcard 35: What is the formula for permutations with repetition?

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

Flashcard 36: What is the number of binary strings of length $5$ with exactly $2$ ones?

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

Flashcard 37: What is the formula for the number of ways to arrange $n$ objects where some are alike?

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

Flashcard 38: How many different 4-digit numbers can be formed using the digits 1-9?

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

Flashcard 39: What is the number of distinct permutations of the letters in $\text{LEVEL}$?

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

Flashcard 40: Identify the number of ways to choose 0 objects from a set of 7.

Answer:

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

Flashcard 41: What is the number of permutations where 3 items are identical and 2 are distinct?

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

Flashcard 42: What is the number of 3-letter strings from $26$ letters if repetition is allowed?

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

Flashcard 43: What is $^5C_0$?

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

Flashcard 44: What is the formula for combinations of $r$ objects from $n$ objects?

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

Flashcard 45: What is the formula for circular permutations of $n$ distinct objects?

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

Flashcard 46: What is the number of 3-letter strings from $26$ letters if repetition is allowed?

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

Flashcard 47: What is the Fundamental Counting Principle for $m$ choices followed by $n$ choices?

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

Flashcard 48: What is the number of permutations of $n$ distinct objects taken all at once?

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

Flashcard 49: What is the value of $1!$?

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

Flashcard 50: What is the Fundamental Counting Principle for $m$ choices followed by $n$ choices?

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

Flashcard 51: What is the number of ways to assign president and vice president from 6 people?

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

Flashcard 52: Identify the rule: when choices are mutually exclusive, how do you count total outcomes?

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

Flashcard 53: What is the number of outcomes when rolling a die and flipping a coin once?

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

Flashcard 54: How many ways can 2 identical balls be placed in 3 different boxes?

Answer:

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

Flashcard 55: How many ways can 4 different books be arranged on a shelf if 2 are always together?

Answer:

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

Flashcard 56: What is the number of ways to assign president and vice president from 6 people?

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

Flashcard 57: What is $^5P_5$?

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

Flashcard 58: What is the number of ways to arrange 8 objects in a circle?

Answer:

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

Flashcard 59: How many distinct permutations can be made from the letters in 'LEVEL'?

Answer:

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

Flashcard 60: How many ways can 8 people sit in a circle?

Answer:

1. Fix one position to eliminate rotational symmetry.

Flashcard 61: How many ways can 2 identical balls be placed in 3 different boxes?

Answer:

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

Flashcard 62: How many ways can the letters in 'ACT' be arranged?

Answer:

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

Flashcard 63: What is the number of 3-letter strings from $26$ letters if repetition is not allowed?

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

Flashcard 64: What is the two-set inclusion–exclusion formula for $|A \cup B|$?

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

Flashcard 65: What is $^5P_5$?

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

Flashcard 66: What does $n!$ represent in counting methods?

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

Flashcard 67: How many ways can a president and vice president be selected from 5 candidates?

Answer:

1. Order matters for president and vice president roles.

Flashcard 68: What is a combination?

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

Flashcard 69: How many ways can 3 identical balls be placed in 4 different boxes?

Answer:

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

Flashcard 70: How many ways can a president and a vice-president be selected from 5 people?

Answer:

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

Flashcard 71: What is the number of distinct permutations of the letters in $\text{MISS}$?

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

Flashcard 72: How many ways can you arrange the letters in the word 'BOOK'?

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

Flashcard 73: What is the formula for permutations of $n$ distinct objects taken $r$ at a time?

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

Flashcard 74: How many distinct permutations can be made from the letters in 'LEVEL'?

Answer:

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

Flashcard 75: In a group of 10 people, how many ways can you choose a committee of 3?

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

Flashcard 76: What is the formula for calculating combinations of $n$ items taken $r$ at a time?

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

Flashcard 77: State the number of ways to arrange 3 books on a shelf.

Answer:

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

Flashcard 78: What is the principle of multiplication in counting?

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

Flashcard 79: How many ways can 3 objects be selected from 5 distinct objects?

Answer:

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

Flashcard 80: What is the formula for the number of subsets of a set with $n$ elements?

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

Flashcard 81: If a lock has 3 dials with 10 digits each, how many combinations are possible?

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

Flashcard 82: What is the number of circular arrangements of $7$ distinct people around a round table?

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

Flashcard 83: How many ways can you arrange the letters in the word 'BOOK'?

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

Flashcard 84: In a group of 10 people, how many ways can you choose a committee of 3?

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

Flashcard 85: What is the number of ways to choose $r$ items from $n$ distinct items when order matters?

Answer: $^nP_r$. Selection where arrangement order matters.

Flashcard 86: What is the formula for calculating permutations of $n$ items taken $r$ at a time?

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

Flashcard 87: What is the factorial definition of $n!$ for a positive integer $n$?

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

Flashcard 88: What is the binomial coefficient notation for combinations?

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

Flashcard 89: What is the formula for the number of subsets of a set with $n$ elements?

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

Flashcard 90: What is the complement rule for counting: how do you count outcomes in set $A$ using $A^c$?

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

Flashcard 91: How many ways can you form a group of 4 from 10 people?

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

Flashcard 92: What is the number of ways to arrange $n$ distinct objects in a circle (rotations considered identical)?

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

Flashcard 93: What is $^5C_5$?

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

Flashcard 94: If a lock has 3 dials with 10 digits each, how many combinations are possible?

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

Flashcard 95: What is the factorial of 5?

Answer:

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

Flashcard 96: Identify the number of ways to arrange 6 people if 2 must sit together.

Answer:

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

Flashcard 97: What is $^8C_2$?

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

Flashcard 98: What is the value of $5!$?

Answer:

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

Flashcard 99: What is $^7P_3$?

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

Flashcard 100: How many different outcomes are possible when tossing 3 coins?

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