# How many 6-digit sequences are ascending?

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.

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.