Capt.Manoj Panday once decided to distribute 180 bullets among his 36 soldiers. But he gave n bullets to a soldier of nth row and there were same number of soldiers in each row. Thus he distributed all his 180 bullets among his soldiers. The number of soldiers in (n - 1)th row was:

Options

A. 3
B. 8
C. 9
D. none of these

Answer

To solve the problem, we need to determine the number of soldiers in the (n - 1)th row given that Captain Manoj Panday distributed 180 bullets among 36 soldiers, with each soldier in the nth row receiving n bullets.

### Step-by-Step Solution:

1. **Understand the total distribution**: We know that there are 180 bullets distributed among 36 soldiers. This means that the total number of soldiers is 36.

2. **Define the number of rows**: Let’s assume there are n rows of soldiers, and each row has the same number of soldiers, which we will denote as x. Therefore, the total number of soldiers can be expressed as:
\[ n \cdot x = 36 \]

3. **Understanding the bullet distribution**: According to the problem, the distribution of bullets is such that:
- The 1st row receives 1 bullet per soldier,
- The 2nd row receives 2 bullets per soldier,
- The 3rd row receives 3 bullets per soldier,
- and so on, up to the nth row, which receives n bullets per soldier.

4. **Calculate total bullets distributed**: The total number of bullets distributed can be expressed as:
\[ \text{Total bullets} = 1 \cdot x + 2 \cdot x + 3 \cdot x + \ldots + n \cdot x \]
This can be simplified using the formula for the sum of the first n natural numbers:
\[ \text{Total bullets} = x \cdot \left( \frac{n(n + 1)}{2} \right) \]
Setting this equal to the total bullets (180):
\[ x \cdot \frac{n(n + 1)}{2} = 180 \]

5. **Substituting for x**: From the equation \( n \cdot x = 36 \), we can express x as:
\[ x = \frac{36}{n} \]
Substituting this into the bullets equation gives:
\[ \frac{36}{n} \cdot \frac{n(n + 1)}{2} = 180 \]

6. **Simplifying the equation**: We can simplify this equation:
\[ 36 \cdot \frac{n + 1}{2} = 180 \]
Multiplying both sides by 2:
\[ 36(n + 1) = 360 \]
Dividing both sides by 36:
\[ n + 1 = 10 \]
Therefore:
\[ n = 9 \]

7. **Finding the number of soldiers in (n - 1)th row**: Since \( n = 9 \), the number of soldiers in each row (x) can be calculated as:
\[ x = \frac{36}{n} = \frac{36}{9} = 4 \]
Thus, the number of soldiers in the (n - 1)th row (which is the 8th row) is also:
\[ x = 4 \]

### Final Answer:
The number of soldiers in the (n - 1)th row is **4**.