# TANCET 2008 Quant - Number Theory, Factors, Permutations

## Permutation Question 20

36 identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least three chairs and there must be at least three rows. A row is parallel to the front of the room. How many different arrangements are possible?

- 2
- 4
- 5
- 6
- 10

**Correct Answer** : 5 arrangements possible. Choice (3)

#### Explanatory Answer

Three conditions have to be satisfied.

1. The number of students per row has to be at least 3.

2. Number of row has to be at least 3.

3. Equal number of students has to be seated in a row.

The following arrangements satisfy all 3 conditions.

Arrangement 1: 3 students to a row; 12 rows.

Arrangement 2: 4 students to a row; 9 rows.

Arrangement 3: 6 students to a row; 6 rows.

Arrangement 4: 9 students to a row; 4 rows.

Arrangement 5: 12 students to a row; 3 rows.

You will observe that the number of students in a row is a factor of 36.

So, an alternative and faster approach is to list down factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

And then start from 3 and quickly find out if the number of rows is at least 2.

Both the conditions are satisfied for the following factors : 3, 4, 6, 9, and 12. i.e., 5 arrangements.

Correct answer choice (3)

Level of difficulty : Moderate