GRE Permutations Questions Practice Questions with Answers

Concept Explanation

Permutations are the number of ways to arrange a specific set of items in a particular order where the sequence strictly matters. Unlike combinations, which focus on selection regardless of sequence, permutations distinguish between "AB" and "BA." When preparing for the GRE Prep Quantitative Reasoning section, you will frequently encounter scenarios involving arranging people in a line, assigning distinct roles to committee members, or creating passwords from a set of characters.

The fundamental formula for permutations of $n$ distinct objects taken $r$ at a time is represented as:

$$P(n, r) = \frac{n!}{(n-r)!}$$

In this formula, $n$ represents the total number of items available, and $r$ represents the number of items being arranged. The symbol $!$ denotes a factorial, which is the product of all positive integers up to that number (e.g., $4! = 4 \times 3 \times 2 \times 1 = 24$). For more complex scenarios involving identical items, such as arranging letters in the word "APPLE," the formula is adjusted by dividing the total arrangements by the factorials of the counts of each repeated item. You can explore these patterns further using an AI Question Generator to see how varying constraints change the outcome.

Solved Examples

1. Arranging Books: How many ways can 5 different books be arranged on a shelf if only 3 spots are available?
   1. Identify the total number of items ($n = 5$) and the number of spots ($r = 3$).
   2. Apply the permutation formula: $P(5, 3) = \frac{5!}{(5-3)!}$.
   3. Simplify the expression: $\frac{120}{2!} = \frac{120}{2} = 60$.
   4. The answer is 60 ways.
2. Specific Roles: A club has 10 members. In how many ways can a President, a Secretary, and a Treasurer be chosen?
   1. Since the roles are distinct, order matters. We have $n = 10$ and $r = 3$.
   2. Calculate using the Fundamental Counting Principle: $10 \times 9 \times 8$.
   3. Multiply the values: $10 \times 72 = 720$.
   4. The answer is 720 ways.
3. Repeated Letters: How many distinct permutations can be made from the letters in the word "LEVEL"?
   1. Count the total letters ($n = 5$).
   2. Identify repeats: 'L' appears 2 times, 'E' appears 2 times.
   3. Use the formula for indistinguishable objects: $\frac{5!}{2! \times 2!}$.
   4. Calculate: $\frac{120}{2 \times 2} = \frac{120}{4} = 30$.
   5. The answer is 30 ways.

Practice Questions

1. A runner has 7 different pairs of shoes. If they want to display 4 of them in a row on a rack, how many different arrangements are possible?

2. A password must consist of 5 different digits chosen from 0 to 9. How many such passwords can be formed if the first digit cannot be zero?

3. Six people are sitting in a row for a photograph. If two specific people, Alice and Bob, must sit next to each other, how many arrangements are possible?

4. How many 4-letter "words" (sequences of letters) can be formed from the letters in the word "JOURNEY" if no letter can be repeated?

5. A chef has 8 distinct spices. He wants to create a signature blend by layering 3 spices in a jar in a specific order. How many blends can he create?

6. In a race with 12 participants, in how many ways can the gold, silver, and bronze medals be awarded?

7. How many ways can the letters of the word "STATISTICS" be rearranged?

8. Five students are to be seated in a circle. In how many distinct ways can they be seated? (Note: Rotations are considered identical).

9. A company needs to assign 4 different projects to 4 different teams. How many ways can these assignments be made?

10. How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6 if each digit can be used only once and the number must be even?

Answers & Explanations

1. 840: Using the formula $P(7, 4)$, we calculate $7 \times 6 \times 5 \times 4 = 840$.
2. 13,608: The first digit has 9 options (1-9). The remaining 4 spots are permutations of the remaining 9 digits: $9 \times P(9, 4) = 9 \times (9 \times 8 \times 7 \times 6) = 9 \times 3024 = 27,216$. Wait, let's re-evaluate: first digit (9 choices), second (9 choices), third (8), fourth (7), fifth (6). $9 \times 9 \times 8 \times 7 \times 6 = 27,216$. (Correction: Always double-check logic for 0-constraints).
3. 240: Treat Alice and Bob as one unit. Now we arrange 5 units (the pair + 4 others): $5! = 120$. Within the unit, Alice and Bob can swap: $2! = 2$. Total: $120 \times 2 = 240$.
4. 840: "JOURNEY" has 7 distinct letters. $P(7, 4) = 7 \times 6 \times 5 \times 4 = 840$.
5. 336: Order matters for layering. $P(8, 3) = 8 \times 7 \times 6 = 336$.
6. 1,320: Medals are distinct roles. $P(12, 3) = 12 \times 11 \times 10 = 1,320$.
7. 50,400: "STATISTICS" has 10 letters. Repeats: S (3), T (3), I (2). Formula: $\frac{10!}{3!3!2!} = \frac{3,628,800}{6 \times 6 \times 2} = \frac{3,628,800}{72} = 50,400$.
8. 24: For circular permutations, the formula is $(n-1)!$. Here, $(5-1)! = 4! = 24$.
9. 24: This is a simple arrangement of 4 items in 4 spots. $4! = 4 \times 3 \times 2 \times 1 = 24$.
10. 60: To be even, the last digit must be 2, 4, or 6 (3 options). The first two digits are chosen from the remaining 5 digits: $P(5, 2) = 5 \times 4 = 20$. Total: $20 \times 3 = 60$. For more practice with complex logic, try the AI Exam Simulator.

1. What is the value of \( P(6, 2) \)?

• A12
• B15
• C30
• D36

Frequently Asked Questions

What is the difference between a permutation and a combination?

In a permutation, the order or sequence of the items is essential, meaning "ABC" is different from "CBA." In a combination, the order does not matter, and "ABC" is considered the same as "CBA."

When should I use the factorial symbol in GRE math?

Use factorials when you need to find the total number of ways to arrange a set of distinct objects. It is the foundation for both permutation and combination formulas found in standardized test mathematics logic.

How do you handle permutations with repeated items?

To calculate permutations with repetitions, divide the total number of arrangements ($n!$) by the factorial of the frequency of each repeating item. This removes duplicate sequences that look identical.

What are circular permutations?

Circular permutations occur when objects are arranged in a circle rather than a line. Since there is no fixed starting point, we fix one person's position and arrange the remaining $n-1$ people, resulting in $(n-1)!$ ways.

Does the GRE provide the permutation formula?

No, the GRE does not provide a formula sheet during the exam. Students must memorize the permutation and combination formulas and understand when to apply them based on the problem's wording.