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Quizzes > High School Quizzes > Mathematics

Permutation or Combination Practice Quiz Worksheet

Practice worksheet featuring permutations and combinations exercises

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting Permutations Combinations Challenge for high school math students.

What is a permutation?
An arrangement of objects where order matters
A selection of objects where order does not matter
A type of mathematical operation unrelated to order
A method to compute combinations
A permutation refers to an arrangement of objects where the order is important. This distinguishes it from a combination where order does not matter.
What is a combination?
An arrangement where order matters
A selection where order does not matter
A mathematical sum of elements
A method for ordering objects
A combination is a selection of items where the order of selection is not important. This is the key difference when compared to a permutation.
How many ways can you arrange 3 distinct books on a shelf?
8
6
9
3
Arranging 3 distinct books is a permutation problem. The number of arrangements is 3! (3 factorial), which equals 6.
How many ways can you choose 2 members from a group of 4 (order does not matter)?
4
12
6
8
Choosing 2 members from 4 without considering the order is a combination problem. The number of combinations is given by C(4,2) = 6.
What does the factorial symbol '!' signify in permutations?
It denotes an exponentiation process
It signifies a subtraction operation performed repeatedly
It represents the product of all positive integers up to that number
It represents the sum of numbers from 1 to n
The factorial symbol '!' indicates the product of all positive integers from 1 to n. This operation is commonly used in permutation calculations.
How many ways can 5 distinct students be seated in a row?
120
100
60
720
Seating 5 distinct students in a row is a permutation problem. The number of possible arrangements is 5! = 120.
In how many ways can you choose a 3-person committee from 7 people?
42
35
21
56
Choosing 3 people from a group of 7 without concern for order is a combination. The number of ways is given by C(7, 3) = 35.
How many 4-digit numbers can be formed using the digits 1, 2, 3, 4 without repetition?
256
24
120
10
Forming a 4-digit number with 4 distinct digits is a permutation problem. The number of arrangements is 4! which equals 24.
How many ways can you arrange 6 different books on a shelf if one specific book must be at the beginning?
24
120
360
720
Fixing one specific book at the beginning leaves you with arranging the remaining 5 books. The number of arrangements for these books is 5!, which equals 120.
How many ways can 3 different prizes be awarded to 8 students if no student wins more than one prize?
504
512
120
336
Awarding 3 prizes to 8 students without repetition is a permutation problem. The number of ways is given by P(8, 3) = 8 � - 7 � - 6 = 336.
In a race with 4 participants, how many different ways can the gold, silver, and bronze medals be awarded?
30
6
24
12
Awarding gold, silver, and bronze medals to 3 out of 4 participants is a permutation problem where order matters. The number of arrangements is P(4, 3) = 24.
How many different 5-letter arrangements can be made from the letters A, B, C, D, E if each letter is used exactly once?
60
120
24
210
Arranging 5 distinct letters is a permutation problem. The total number of arrangements is 5! which equals 120.
If a combination lock uses 3 distinct digits chosen from 0 to 9, how many possible combinations exist assuming order matters?
840
720
360
900
Since order matters for the lock, it is a permutation problem. With 10 digits and no repetition, the number of arrangements is given by P(10, 3) = 10 � - 9 � - 8 = 720.
How many committees of 2 can be formed from 6 people if the order does not matter?
15
30
12
20
This is a combination problem because the order in which the committee members are chosen does not matter. The calculation C(6, 2) yields 15 possible committees.
How many distinct arrangements are there for arranging 5 flags on a pole if all flags are distinct?
130
120
110
100
Arranging 5 distinct flags is a permutation problem since the order matters. The number of arrangements is 5! which equals 120.
How many distinct arrangements can be made of the letters in the word 'BANANA'?
60
90
120
30
The word 'BANANA' has 6 letters with repeating characters: A appears 3 times, N appears 2 times, and B appears once. Using the formula for permutations of multiset, the number of distinct arrangements is 6!/(3!� - 2!) = 60.
A round table has 8 seats. How many distinct ways can 8 people be seated around the table if rotations are considered identical?
5040
840
40320
720
When seating people around a circular table, one position is fixed to remove rotational symmetry. This leaves (8-1)! or 7! = 5040 distinct arrangements.
In a class of 12 students, if 2 specific students cannot be on the same committee, how many 4-person committees can be formed?
495
460
405
450
First, calculate the total number of committees using C(12, 4) which is 495. Then subtract the committees that include both specific students, which is C(10, 2) = 45, leaving 495 - 45 = 450 committees.
How many different 7-digit numbers can be formed using digits 1 through 9, if no digit is repeated and the number must be odd?
80640
100800
90720
120960
An odd number requires its last digit to be one of the 5 odd digits (1, 3, 5, 7, 9). The remaining 6 digits are a permutation of 8 available digits. Thus, the total number is 5 � - P(8,6) = 5 � - 20160 = 100800.
How many ways can you distribute 5 different tasks among 4 students if each student must get at least one task?
240
600
360
480
This problem requires you to assign distinct tasks such that every student gets at least one. One approach is to determine which student gets 2 tasks (4 choices), select 2 tasks for that student (C(5, 2) = 10), and then assign the remaining 3 tasks to the other 3 students in 3! = 6 ways. Multiplying these together gives 4 � - 10 � - 6 = 240.
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Study Outcomes

  1. Understand fundamental principles of permutations and combinations.
  2. Analyze problem statements to determine appropriate combinatorial strategies.
  3. Apply formulas to compute arrangements and selections accurately.
  4. Evaluate results to verify solutions in practice quiz scenarios.
  5. Synthesize problem-solving steps to tackle complex combinatorial challenges.

Permutations & Combinations Worksheet Cheat Sheet

  1. Understand the difference between permutations and combinations - Permutations take order seriously, while combinations throw order out the window. Grasping this key distinction will help you decide which formula to use in any counting problem. Britannica: Permutation Basics
  2. Master the factorial function (n!) - The factorial of n multiplies all positive integers up to n (n × (n−1) × … × 1). This concept is the building block for both permutation and combination formulas, powering your counting skills. Math is Fun: Combinatorics Overview
  3. Learn the formula for permutations without repetition - P(n, r) = n! / (n−r)! counts how many ways you can arrange r items out of n when order matters. This formula is perfect for lineups, seat assignments, and password arrangements. Britannica: Permutation Formula
  4. Understand the formula for combinations without repetition - C(n, r) = n! / (r! × (n−r)!) finds how many groups of r you can pick from n when order doesn't matter. Use this for team selections, handshakes, or picking lottery numbers. Britannica: Combination Formula
  5. Practice distinguishing between permutations and combinations - Build intuition by labeling scenarios as "order matters" (permutations) or "order doesn't matter" (combinations). With a little practice, you'll instantly identify which counting tool to grab. GeeksforGeeks: Perm vs Comb
  6. Explore real-life applications - From calculating lottery odds to arranging family photos, permutations and combinations are everywhere. Linking abstract formulas to real-life puzzles makes them stickier in your memory. GeeksforGeeks: Practical Uses
  7. Familiarize yourself with repetition cases - Sometimes you can pick the same item more than once (like choosing ice-cream scoops!). Special formulas handle these situations, so you won't get tripped up on repeats. Math is Fun: Repetition Counts
  8. Use fun mnemonics - Remember "P for Position" to cue that permutations involve order, while combinations skip the sequence. Mnemonics turn tricky definitions into brain-friendly memory hooks. GeeksforGeeks: Mnemonics
  9. Practice with varied constraints - Tackle problems with limits like "choose at least two" or "no repeats allowed" to deepen your flexibility. Facing different rules makes you a more agile problem-solver. GeeksforGeeks: Practice Problems
  10. Review derivations for deeper understanding - Instead of just memorizing formulas, follow the step-by-step derivations. Connecting each line of algebra to a counting concept cements the logic in your mind. GeeksforGeeks: Formula Derivation
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