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Quizzes > Mathematics > Advanced Math

Counting Principle and Permutations Quiz: Test Your Skills Now

Take the Counting Principle Calculator Challenge - Dive into Permutation Practice

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration of quiz scene on coral background with calculator numbers shapes and counting principle symbols

Use this Fundamental Counting Principle Calculator Quiz to practice counting outcomes for simple and multi-step events, including basic permutations. Work through quick questions and see instant answers to spot gaps before a test. Want a fresh warm-up first? Try another math quiz .

If there are 3 shirt options and 2 pant options, how many unique outfits can be formed?
3
6
5
9
The Fundamental Counting Principle states that when two independent choices are made, the total number of outcomes is the product of the choices. Here, 3 shirt choices multiplied by 2 pant choices equals 6. This approach is detailed in the multiplication principle.
A license plate consists of three letters followed by two digits. How many different license plates are possible if repetition is allowed?
26 + 10 = 36
26^3 × 10^2 = 1,757,600
26^3 + 10^2 = 17576 + 100
3! × 2! = 12
With repetition allowed, each of the three letter slots has 26 possibilities and each of the two digit slots has 10 possibilities. By the Fundamental Counting Principle, multiply 26^3 by 10^2 to get 1,757,600. This method is covered here.
How many passwords can be created using exactly four digits followed by two letters, if characters can repeat?
10^4 × 26^2 = 6,760,000
4! × 2! = 48
10^2 × 26^4 = 4,569,760
10^4 + 26^2 = 10,676
Since digits can repeat, each of the four digit positions has 10 options, and each of the two letter positions has 26 options. The total number is 10^4 × 26^2 = 6,760,000 by the product rule. This principle is explained here.
When rolling two six-sided dice, how many possible outcomes exist?
36
18
6
12
Each die has 6 possible results, and the outcomes are independent. By multiplying, 6 × 6 = 36 total outcomes. This is a classic example of the Fundamental Counting Principle.
A game involves flipping two coins and rolling one die. How many total outcome combinations are there?
2 × 2 + 6 = 10
2^2 + 6 = 10
2 + 2 + 6 = 10
2^2 × 6 = 24
Each coin has 2 outcomes and the die has 6 outcomes. By the product rule, multiply 2 × 2 × 6 = 24 possible results. This is directly from the Fundamental Counting Principle.
If a snack machine has 4 snack choices and a drink machine has 3 drink choices, how many snack-drink pairs can be chosen?
4 + 3 = 7
3^4 = 81
4^3 = 64
4 × 3 = 12
Each snack can pair with each drink independently. Multiply 4 options by 3 options to get 12 combinations. This uses the Fundamental Counting Principle.
A weekly planner has 7 days and each day has 3 meal slots. How many day-meal combinations exist?
3^7 = 2187
7 × 3 = 21
7 + 3 = 10
7^3 = 343
Choosing a day and then a meal slot are independent choices. Multiply 7 days by 3 meals to get 21 possible combinations. This illustrates the basic product rule.
In how many ways can 4 different books be arranged on a shelf?
4 + 4 = 8
4^4 = 256
4! = 24
4 × 4 = 16
Arranging 4 distinct items in order is a permutation count given by 4! = 4×3×2×1 = 24. Each new position has one fewer choice. This is an extension of the multiplication principle.
How many four-digit PIN codes are possible if digits cannot repeat and leading zeros are allowed?
4! = 24
9 × 9 × 8 × 7 = 4536
10^4 = 10000
10 × 9 × 8 × 7 = 5040
For each PIN, the first digit has 10 choices, the next has 9 remaining, then 8, then 7. Multiply 10×9×8×7 = 5040. Leading zeros are treated like any other digit.
A restaurant offers 3 appetizers, 5 entrees, and 2 desserts. How many three-course meals can a customer choose?
3 + 5 + 2 = 10
3 × 5 + 2 = 17
3 × 5 × 2 = 30
3^5 × 2 = 486
Each course choice is independent, so multiply 3 appetizers by 5 entrees by 2 desserts to get 30 combinations. This is a direct use of the counting principle.
A license plate has two letters followed by three digits, and both letters and digits may repeat. How many plates are possible?
2! + 3! = 8
26^2 + 10^3 = 676 + 1000
26P2 × 10P3 = 15600
26^2 × 10^3 = 676000
With repetition allowed, each letter slot has 26 options and each digit slot has 10. Multiply 26^2 by 10^3 to get 676,000 possible plates. This follows the multiplication rule.
From 10 candidates, how many ways can you choose a president, vice president, and treasurer (no one can hold more than one office)?
10! = 3,628,800
10^3 = 1000
10 + 9 + 8 = 27
10 × 9 × 8 = 720
For each office you choose from the remaining pool: 10 choices for president, then 9 for vice, then 8 for treasurer. Multiply to get 720. This is a permutation application of FCP.
How many possible outcomes are there when flipping four coins?
2^4 = 16
4 × 2 = 8
4 + 2 = 6
2 × 4! = 48
Each coin flip has 2 outcomes and flips are independent. Multiply 2×2×2×2 = 16. This is a simple application of the Fundamental Counting Principle.
A password consists of two uppercase letters, two lowercase letters, and two digits (all can repeat). How many possible passwords exist?
26^2 × 26^2 × 10^2 = 45,697,600
26^4 + 10^2 = 45,697,600
62^6 = 56,800,235,584
52^4 × 10^2 = 7,311,616,000
Each uppercase slot has 26 choices, each lowercase slot has 26, and each digit slot has 10. Multiply these: 26^4×10^2 = 45,697,600. This extends the basic product rule to multiple categories.
How many distinct arrangements (anagrams) are there of the letters in the word MISSISSIPPI?
11P11 = 39,916,800
11! = 39,916,800
11!/(4!4!2!) = 34,650
4!4!2! = 192
The word has 11 letters with repeats: 4 S's, 4 I's, and 2 P's. The count is 11! divided by the factorial of each repeated group: 11!/(4!4!2!) = 34,650. This generalizes FCP with division for identical items.
A license plate has three distinct letters (no repeats) followed by three digits (repeats allowed). How many plates are possible?
26C3 × 10^3 = 2,600,000
26^3 × 10^3 = 17,576,000
26P3 × 10^3 = 15,600,000
26P3 + 10P3 = 15,600
Choose 3 distinct letters in order: 26P3 = 26×25×24, and each of the three digits has 10 choices. Multiply 26P3 by 10^3 = 15,600,000. This applies permutations then the multiplication principle.
How many 5-digit even numbers have no repeating digits?
9 × 8 × 7 × 6 × 5 = 15,120
13,776
5 × 4 × 3 × 2 × 1 = 120
10P5 = 30,240
Casework: if the first digit is odd (5 choices) and last even (5), middle 3 from 8 remaining: 5×5×(8×7×6)=8400; if first even (4 choices) and last even from 4, middle 3: 4×4×(8×7×6)=5376. Sum = 13,776. This uses FCP with careful enumeration.
In how many ways can 8 people be seated around a round table so that two specific people are not next to each other?
5040 - 1440 = 3600
6! = 720
(8-1)! = 5040
7! × 2 = 10080
Total circular arrangements of 8 is (8?1)! = 5040. Count those where the two sit together by treating them as one unit: (7?1)!×2 = 1440. Subtract: 5040?1440 = 3600. This uses the product rule and inclusion - exclusion.
How many ways can 4 distinct tasks be assigned to 6 workers if each worker can do at most one task?
6^4 = 1296
4^6 = 4096
6P4 = 360
6C4 = 15
You choose a worker for each task without repetition: 6 options for the first task, 5 for the second, then 4 and 3. Multiply to get 6P4 = 360. This applies permutations and FCP.
A combination lock has 3 dials labeled 0 - 9 and no digit may repeat. How many combinations can be set?
10P10 = 10!
3 × 10 = 30
10^3 = 1000
10 × 9 × 8 = 720
Each of the three positions has one fewer option than the last due to no repetition. Multiply 10×9×8 = 720. This is a standard FCP example.
How many distinct 6-character codes are there of the form letter-letter-digit-digit-letter-letter, where digits cannot repeat but letters can?
26P4 × 10^2 = 3,237,360
26^2 × 10P4 = 3,276,480
26^4 × 10P2 = 41,127,840
26^4 × 10^2 = 45,697,600
There are 26 choices for each letter slot (4 total), and for the two digit slots without repetition: 10×9. Multiply: 26^4 × (10P2) = 41,127,840. This uses FCP with permutations for digits.
How many nonnegative integer solutions exist for x? + x? + x? = 10?
C(10,3) = 120
10P3 = 720
C(12,2) = 66
10^3 = 1000
This is a stars-and-bars problem equivalent to choosing 2 dividers among 12 slots: C(10+3?1,3?1) = C(12,2) = 66. It combines combinatorial reasoning with FCP.
How many distinct 7-letter 'words' can be formed using exactly three A's and four B's?
P(7,3) = 210
3^4 + 4^3 = 81
C(7,3) = 35
7! = 5040
The number of ways to place 3 A's in 7 positions is C(7,3); the B's fill the rest. That equals 35. This uses combinations and the product rule conceptually.
A phone number has seven digits. How many numbers have at least one repeated digit?
10^7 = 10,000,000
10P7 = 6048000
10P7 ? 10^7 = ?3,952,000
10^7 ? 10P7 = 3,952,000
Total phone numbers: 10^7. Those with no repeats: P(10,7)=6,048,000. Subtract to get 10,000,000 ? 6,048,000 = 3,952,000 with at least one repeat. This combines complement counting with FCP.
In a race of 8 runners, medals are awarded to the top three finishers (distinct positions). How many possible outcomes are there?
8! = 40,320
8^3 = 512
P(8,3) = 336
C(8,3) = 56
Assigning gold, silver, and bronze is an ordered selection of 3 from 8, which is P(8,3) = 8×7×6 = 336. This is a direct permutation application of FCP.
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Study Outcomes

  1. Understand the Fundamental Counting Principle -

    Explain the core rule that the total number of possible outcomes is the product of individual event choices and illustrate its significance in counting problems.

  2. Apply the Fundamental Counting Principle Calculator -

    Use the interactive fundamental counting principle calculator to break down complex multi-step scenarios into manageable calculations for accurate results.

  3. Use the Permutation Calculator -

    Compute ordered arrangements efficiently by leveraging the permutation calculator and distinguish when order matters in different counting contexts.

  4. Analyze Counting Scenarios -

    Determine whether to apply the fundamental counting principle, permutations, or combinations based on scenario requirements and constraints.

  5. Solve Permutation Practice Problems -

    Engage with a variety of permutation practice problems to reinforce concepts and receive instant feedback through the fundamental counting principle quiz.

  6. Evaluate and Verify Outcomes -

    Interpret quiz results and use the counting principle calculator to cross-check solutions, ensuring confidence in problem-solving accuracy.

Cheat Sheet

  1. Multiply Choices - The Fundamental Counting Principle -

    When a process has sequential steps with m options for the first and n for the second, total outcomes = m×n, extending to more steps as m×n×p… (MIT OpenCourseWare). A handy mnemonic is "Multiply Your Choices" to remember this rule. Use the fundamental counting principle calculator to verify your manual computations instantly.

  2. Permutations - Ordered Arrangements with P(n, r) -

    Permutations count ordered selections and follow the formula P(n, r)=n!/(n−r)!; for example, P(5,3)=5×4×3=60 (Khan Academy). This highlights how order multiplies your options when arranging distinct items. Practice with a permutation calculator and permutation practice problems to solidify your understanding.

  3. Permutations with Repetition - Identical Items Counted -

    When some items repeat, the permutations formula becomes n!/(n1! n2!…) to account for indistinguishable elements, such as arranging the letters in "BALLOON" (University of Cambridge). This adjustment avoids overcounting and keeps calculations accurate. A counting principle calculator can help confirm your results for complex letter or digit arrangements.

  4. Combinations vs. Permutations - Order Doesn't Always Matter -

    Combinations disregard order and use C(n,r)=n!/(r!(n−r)!), e.g. selecting 2 fruits from 5 gives C(5,2)=10 unique pairs (Harvard University materials). Recognizing when order is irrelevant is key to choosing the right formula. Complement your studies with fundamental counting principle quiz questions to test this distinction.

  5. Tree Diagrams & Calculator Strategy - Visual and Digital Tools -

    Tree diagrams map each decision path visually, reinforcing the fundamental counting principle before you turn to a digital counting principle calculator. Combining sketches with an online fundamental counting principle calculator or permutation calculator boosts accuracy and confidence. This two-step approach - visual then digital - ensures you master both intuitive and computational methods.

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