Easy GRE Combinations Questions Practice Questions

Concept Explanation

Combinations represent the number of ways to select a subset of items from a larger group where the order of selection does not matter.

In the context of GRE Prep, distinguishing between combinations and permutations is essential. While permutations focus on arrangements (where order is important, like a race finish or a password), combinations focus solely on the group of items chosen. For example, selecting a committee of three people from a group of ten is a combination problem because the group is the same regardless of who was picked first, second, or third.

The standard formula for calculating combinations is expressed as "n choose k," denoted as $C(n, k)$ or $\binom{n}{k}$:

$$C(n, k) = \frac{n!}{k!(n-k)!}$$

In this formula:

• n is the total number of items in the set.
• k is the number of items being chosen.
• ! denotes a factorial, which is the product of an integer and all the integers below it (e.g., $4! = 4 \times 3 \times 2 \times 1 = 24$).

A helpful shortcut for Easy GRE Combinations Questions is to use the "countdown" method. To find $C(8, 3)$, start at 8 and write the first 3 terms of the factorial in the numerator, then put $3!$ in the denominator: $\frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$. This method avoids calculating large factorials and simplifies quickly. You can find more foundational math tips in our Free GRE Practice Questions guide.

Solved Examples

1. Example 1: Basic Selection
   How many different groups of 2 students can be chosen from a class of 5 students?
   1. Identify $n = 5$ and $k = 2$.
   2. Apply the combination formula: $\frac{5!}{2!(5-2)!}$.
   3. Simplify: $\frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10$.
   4. There are 10 unique groups possible.
2. Example 2: Choosing Toppings
   A pizza parlor offers 6 different toppings. If a customer wants to choose exactly 3 toppings, how many different combinations are available?
   1. Identify $n = 6$ and $k = 3$.
   2. Set up the calculation: $\frac{6 \times 5 \times 4}{3 \times 2 \times 1}$.
   3. Cancel the $3 \times 2$ in the denominator with the 6 in the numerator.
   4. Calculate the remaining values: $5 \times 4 = 20$.
3. Example 3: Committee Formation
   A manager must select 4 employees from a team of 7 to attend a conference. How many different sets of employees can be selected?
   1. Identify $n = 7$ and $k = 4$.
   2. Note that $C(n, k)$ is equal to $C(n, n-k)$. Thus, $C(7, 4) = C(7, 3)$.
   3. Calculate $C(7, 3)$: $\frac{7 \times 6 \times 5}{3 \times 2 \times 1}$.
   4. Simplify: $7 \times 5 = 35$.

Practice Questions

1. A book club has 8 members. In how many ways can a committee of 2 members be chosen to pick the next book?
2. A fruit basket contains an apple, an orange, a pear, a banana, and a grape. If you choose 3 pieces of fruit, how many different combinations can you make?
3. There are 10 points on a circle. How many different triangles can be formed using these points as vertices?

4. A researcher needs to select 3 volunteers from a pool of 9 candidates. How many different groups of volunteers are possible?
5. In a deck of 12 unique cards, how many ways can a player be dealt a hand of 5 cards?
6. A small company has 6 employees. If they need to form a task force of 4 people, how many different task forces can be created?
7. How many ways can a student choose 2 electives from a list of 11 available courses?
8. A jewelry maker has 5 different gemstones. How many ways can they choose 4 gemstones to set in a bracelet?
9. A coach must choose 5 starters from a basketball team of 12 players. How many different starting lineups can be formed (assuming positions don't matter)?
10. A florist has 7 types of flowers. A customer wants a bouquet consisting of 5 different types of flowers. How many combinations are possible?

Answers & Explanations

1. 28. Use $C(8, 2)$. Calculation: $\frac{8 \times 7}{2 \times 1} = 28$. Since the order of the committee members doesn't matter, it is a combination.
2. 10. Use $C(5, 3)$, which is the same as $C(5, 2)$. Calculation: $\frac{5 \times 4}{2 \times 1} = 10$.
3. 120. To form a triangle, you must choose 3 points out of 10. Use $C(10, 3)$. Calculation: $\frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$.
4. 84. Use $C(9, 3)$. Calculation: $\frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 3 \times 4 \times 7 = 84$.
5. 792. Use $C(12, 5)$. Calculation: $\frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1}$. After canceling, you get $11 \times 9 \times 8 = 792$. You can use the AI Exam Simulator to practice similar high-volume calculations under timed conditions.
6. 15. Use $C(6, 4)$, which is equal to $C(6, 2)$. Calculation: $\frac{6 \times 5}{2 \times 1} = 15$.
7. 55. Use $C(11, 2)$. Calculation: $\frac{11 \times 10}{2 \times 1} = 11 \times 5 = 55$.
8. 5. Use $C(5, 4)$, which is equal to $C(5, 1)$. Any set of "n choose n-1" is always simply $n$. Thus, the answer is 5.
9. 792. Use $C(12, 5)$. As calculated in question 5, the result is 792.
10. 21. Use $C(7, 5)$, which is equal to $C(7, 2)$. Calculation: $\frac{7 \times 6}{2 \times 1} = 21$. For more practice with varied sets, check out our GRE Practice Questions with Explanations.

1. Which of the following scenarios describes a combination?

• AArranging 5 books on a shelf in a specific order
• BAssigning 1st, 2nd, and 3rd place finishes in a race
• CChoosing a group of 3 friends to go to the movies
• DSelecting a 4-digit PIN for a debit card

Frequently Asked Questions

What is the main difference between combinations and permutations?

The primary difference is whether order matters. Permutations are used for arrangements where the sequence is important, while combinations are used for selections where the group remains the same regardless of order.

Why is $C(n, k)$ the same as $C(n, n-k)$?

Choosing $k$ items to include in a group is mathematically equivalent to choosing $n-k$ items to exclude from that group. This symmetry makes calculations easier for large values of $k$.

Does the GRE provide the combination formula on screen?

No, the GRE does not provide formulas during the exam. Students must memorize the combination and permutation formulas as part of their preparation for the Quantitative Reasoning section.

Can I use a calculator for combinations on the GRE?

Yes, the GRE provides an on-screen calculator. However, it is often faster to simplify the combination formula manually using the countdown method than to type out large factorials.

What are some common keywords for combination problems?

Look for words like "select," "group," "committee," "set," or "choose." If the problem does not imply a specific ranking or sequence, it is likely a combination question.