How many 5 digits numbers exist such that the sum of digits is equal to 10?

We need to determine how many 5-digit numbers exist such that the sum of their digits equals 10.

Let’s represent the 5-digit number as \( d_1 d_2 d_3 d_4 d_5 \), where \( d_1, d_2, d_3, d_4, d_5 \) are digits, and each \( d_i \) (except \( d_1 \)) can range from 0 to 9. However, \( d_1 \) (the first digit) must be between 1 and 9, as the number is a 5-digit number.

Thus, the equation we need to solve is:
\[
d_1 + d_2 + d_3 + d_4 + d_5 = 10
\]
where \( 1 \leq d_1 \leq 9 \) and \( 0 \leq d_2, d_3, d_4, d_5 \leq 9 \).

### Transformation:
To simplify, let’s define a new variable \( d_1' = d_1 - 1 \), where \( d_1' \geq 0 \). This transforms the equation to:
\[
d_1' + d_2 + d_3 + d_4 + d_5 = 9
\]
Now, \( d_1', d_2, d_3, d_4, d_5 \geq 0 \).

This is a classic stars and bars problem, where we need to find the number of non-negative integer solutions to the equation above.

The number of solutions is given by the formula for combinations with repetition:
\[
\binom{9 + 5 - 1}{5 - 1} = \binom{13}{4}
\]

Calculating \( \binom{13}{4} \):
\[
\binom{13}{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715
\]

Thus, there are **715** different 5-digit numbers such that the sum of their digits equals 10.

Would you like further explanation or have any questions on this? Here are some related questions you might find interesting:

1. How many 4-digit numbers exist such that the sum of digits equals 9?
2. How many 6-digit numbers exist such that the sum of digits equals 15?
3. What is the number of solutions to the equation \(x_1 + x_2 + x_3 = 20\), where \(0 \leq x_1, x_2, x_3 \leq 10\)?
4. What if the sum of digits had to be 12 instead of 10?
5. How many ways can we distribute 10 identical objects into 5 distinct boxes?

**Tip**: In combinatorics, "stars and bars" is a powerful tool for finding the number of ways to distribute identical objects into distinct groups.

Explore how to find the number of 5-digit numbers where the sum of digits equals 10 using combinatorics and the stars and bars theorem. Learn step-by-step with examples, suitable for Grades 10-12 students.

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