GRE® Permutation Combination & Probability Q2

This GRE quant practice question is a permutation combination problem solving question. A classic example of sampling without replacement and where the order does NOT matter. An ⁿC_r question.

Question 2: In how many ways can 3 students be selected from a group of 12 students to represent a school in the inter school essay competition?

1. 33
2. 12!
3. 1320
4. 220
5. 36

---

---

---

---

Explanatory Answer | GRE Permuation Combination Question 2

Step 1 of solving this GRE Permutation Question: Is this an example of sampling with replacement or without replacement?

What kind of sampling?
Total number of students 12. Students to be selected 3.
We need to select 3 students from a class of 12 students.
The first student can be any of the 12 students.
The second student to be selected will come from a reduced set of 11 students.
Else, there are instances where the student selected first could be selected again - in that case we will not have selected 3 students.
Hence, sampling without replacement

Once, we have figured that it is an example of sampling without replacement, select 3 students out of 12 in ¹²C₃ ways.

Step 2 of solving this GRE Permutation Combination Question: Is it an example of sampling where the order matters or one in which the order does not matter?

Let us say students A, B, and C have been selected from among the 12.
Is it different from saying students B, C, and A have been selected?
Order does not matter
If order does not matter, you should not multiply the answer in the previous step with the number of ways things can be reordered.
So, the answer is ¹²C₃
¹²C₃ = \\frac{12!}{(12 - 3)! × 3!}) = \\frac{12!}{9! × 3!}) = \\frac{12 × 11 × 10 × 9!}{9! × 3 × 2 × 1}) = \\frac{12 × 11 × 10}{3 × 2 × 1} ) = 220 ways.

Choice D is the correct answer