A Pascal-like triangle has $10$ as the top row and $10$ followed by $1$ as the second row. In each subsequent row the first number is $10$ , the last number is $1$ , and, as in the standard Pascal's Triangle, each other number in the row is the sum of the two numbers directly above it. The first four rows are shown below:

$\begin{array}{ccccccc} &&&10&&&\\ &&10&&1&& \\ &10&&11&&1& \\ 10&&21&&12&&1 \\ \end{array}$

What is the sum of the digits of the sum of the numbers in the $11$ th row?