A rectangle has a perimeter of 26 . How many combinations of integral valued length are possible ?

Options

A. 4
B. 8
C. 6
D. 12

Answer

To find the number of combinations of integral valued lengths for a rectangle with a given perimeter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Perimeter Formula**:
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2 \times (length + breadth) \]
Given that the perimeter is 26, we can set up the equation:
\[ 2 \times (L + B) = 26 \]

2. **Simplify the Equation**:
Dividing both sides of the equation by 2 gives:
\[ L + B = 13 \]
Here, L represents the length and B represents the breadth of the rectangle.

3. **Express Breadth in Terms of Length**:
We can express breadth (B) in terms of length (L):
\[ B = 13 - L \]

4. **Determine Integral Values**:
Since both L and B must be positive integers, we need to find the values of L such that B remains positive:
\[ 13 - L > 0 \implies L < 13 \]
Therefore, the possible integral values for L are from 1 to 12 (inclusive).

5. **Count the Combinations**:
The integral values for L can be:
- If \( L = 1 \), then \( B = 12 \)
- If \( L = 2 \), then \( B = 11 \)
- If \( L = 3 \), then \( B = 10 \)
- If \( L = 4 \), then \( B = 9 \)
- If \( L = 5 \), then \( B = 8 \)
- If \( L = 6 \), then \( B = 7 \)
- If \( L = 7 \), then \( B = 6 \)
- If \( L = 8 \), then \( B = 5 \)
- If \( L = 9 \), then \( B = 4 \)
- If \( L = 10 \), then \( B = 3 \)
- If \( L = 11 \), then \( B = 2 \)
- If \( L = 12 \), then \( B = 1 \)

This gives us a total of 12 possible integral values for length (L).

### Conclusion:
The number of combinations of integral valued lengths possible for the rectangle is **12**.