Lesson 15

For numbers where the digits decrease, we can correspond sets of seven digits to them. For example, the number \(9765210\) corresponds to the set \(\{9,7,6,5,2,1,0\}\). Conversely, the set \(\{4, 2, 0, 6, 8, 7, 3\}\) corresponds to the number \(8764320\). However, if the digits must not increase, such a correspondence will not be a bijection because digits can repeat. We only need to know how many times each of the ten digits repeats. Thus, the desired number is the number of solutions to the equation \[x_{9}+x_{8}+\dotsc+x_{0}=7\] where \(x_{9}, x_{8}, \dotsc, x_{0} \in \mathbb{Z}_{\ge 0}\). We will soon find these two quantities.