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 ^{n}C_{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?

- 33
- 12!
- 1320
- 220
- 36

From INR

**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 ^{12}C_{3} ways.**

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 ^{12}C_{3}

^{12}C_{3} = \\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.**

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