[Math] How many 6-digit sequences are ascending

Question

How many 6-digit sequences are ascending, like 023689, 033588, or 222222?

A number may begin with 0 and can be repeated, but must be increasing.

I am not entirely certain how to approach this problem.

Answer 1

Notice that since the digits in the string are ascending, such a number is completely determined by how many times each digit appears in the string. For example, if the six digits $0, 2, 3, 6, 8, 9$ each appear once, we obtain the string $023689$. If the digits $0$ and $5$ each appear once and the digits $3$ and $8$ each appear twice in the six-digit string, then we obtain the string $033588$.

Let $x_i$, $0 \leq i \leq 9$, be the number of times that the digit $i$ appears in the six-digit string. Then $$x_0 + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 = 6$$ is an equation in the nonnegative integers. The number of ascending strings of length six is the number of solutions of this equation in the nonnegative integers.

A particular solution of the equation $$x_0 + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 = 6$$ corresponds to the placement of $10 - 1 = 9$ addition signs in a row of six ones. For instance, $$1 + + 1 + 1 + + + 1 + + 1 + 1$$ corresponds to the solution $x_0 = 1, x_1 = 0, x_2 = x_3 = 1, x_4 = x_5 = 0, x_6 = 1, x_7 = 0, x_8 = x_9 = 1$ and the string $023689$, while $$1 + + + 1 1 + + 1 + + + 1 1 +$$ corresponds to the solution $x_0 = 1, x_1 = x_2 = 0, x_3 = 2, x_4 = 0, x_5 = 1, x_6 = x_7 = 0, x_8 = 2, x_9 = 0$ and the string $033588$. The number of such solutions is $$\binom{6 + 10 - 1}{10 - 1} = \binom{15}{9}$$ since we must choose which nine of the fifteen symbols required for six ones and nine addition signs will be filled with addition signs.