In a school hall, 460 students were sitting in rows and columns in such a way that the number of students sitting in each column was three more than the number of students sitting in each row.The number of students in each column was:

Options

A. 0
B. 23
C. 24
D. None of these

Answer

To solve the problem, we need to set up equations based on the information given. Let's denote:

- Let \( r \) be the number of students sitting in each row.
- Let \( c \) be the number of students sitting in each column.

According to the problem, we have two key pieces of information:

1. The total number of students is 460.
2. The number of students in each column is 3 more than the number of students in each row.

From this information, we can set up the following equations:

1. The total number of students can be expressed as:
\[ r \times c = 460 \]

2. The relationship between the number of students in rows and columns is:
\[ c = r + 3 \]

Now, we can substitute the second equation into the first equation.

### Step 1: Substitute \( c \) in the total students equation
Substituting \( c \) in the equation \( r \times c = 460 \):
\[ r \times (r + 3) = 460 \]

### Step 2: Expand the equation
Expanding the equation gives us:
\[ r^2 + 3r = 460 \]

### Step 3: Rearrange the equation
Rearranging the equation to set it to zero:
\[ r^2 + 3r - 460 = 0 \]

### Step 4: Solve the quadratic equation
Now, we can solve this quadratic equation using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 3 \), and \( c = -460 \).

Calculating the discriminant:
\[ b^2 - 4ac = 3^2 - 4 \times 1 \times (-460) = 9 + 1840 = 1849 \]

Now, applying the quadratic formula:
\[ r = \frac{-3 \pm \sqrt{1849}}{2 \times 1} \]
Calculating \( \sqrt{1849} = 43 \):
\[ r = \frac{-3 \pm 43}{2} \]

Calculating the two possible values for \( r \):
1. \( r = \frac{40}{2} = 20 \)
2. \( r = \frac{-46}{2} = -23 \) (not a valid solution since the number of students cannot be negative)

Thus, \( r = 20 \).

### Step 5: Find \( c \)
Now, substituting \( r \) back into the equation for \( c \):
\[ c = r + 3 = 20 + 3 = 23 \]

### Conclusion
The number of students in each column is \( \boxed{23} \).