Number of four digit number between `7000` and `8000`, such that the thousands digit is equal to the sum of other three digit, is

Options

A. `36`
B. `38`
C. `15`
D. `18`

Explanation

To solve the problem of finding the number of four-digit numbers between 7000 and 8000 such that the thousands digit equals the sum of the other three digits, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range**:
- The four-digit numbers between 7000 and 8000 have their thousands digit fixed as 7.
- Therefore, we can represent any number in this range as \( 7xyz \), where \( x, y, z \) are the digits in the hundreds, tens, and units places respectively.

2. **Set Up the Equation**:
- According to the problem, the thousands digit (which is 7) must equal the sum of the other three digits.
- This gives us the equation:
\[ x + y + z = 7 \]

3. **Determine the Constraints**:
- The digits \( x, y, z \) can take values from 0 to 9. However, since we are summing to 7, they will naturally be constrained to non-negative integers.

4. **Use the Stars and Bars Method**:
- To find the number of non-negative integer solutions to the equation \( x + y + z = 7 \), we can use the "stars and bars" combinatorial method.
- The formula for the number of solutions in non-negative integers to the equation \( x_1 + x_2 + ... + x_r = n \) is given by:
\[ \binom{n + r - 1}{r - 1} \]
- Here, \( n = 7 \) (the total sum) and \( r = 3 \) (the number of variables: \( x, y, z \)).

5. **Calculate the Combinations**:
- Plugging in the values into the formula:
\[ \binom{7 + 3 - 1}{3 - 1} = \binom{9}{2} \]
- Now, calculate \( \binom{9}{2} \):
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \]

6. **Conclusion**:
- Therefore, the number of four-digit numbers between 7000 and 8000 such that the thousands digit equals the sum of the other three digits is **36**.

### Final Answer:
The number of four-digit numbers between 7000 and 8000 such that the thousands digit is equal to the sum of the other three digits is **36**.