How many 5 digits numbers are there, whose digits sum to 22? 
How many 5 digits numbers are there, whose digits sum to 22? Of course the first digit has to be larger than 0.

Using generating functions:
$$[x^{22}](x+x^2+\cdots+x^9)(1+x+\cdots +x^9)^4=\\
[x^{21}]\frac{1-x^9}{1-x}\cdot \frac{(1-x^8)^4}{(1-x)^4}=\\
[x^{21}](1-x^9)(1-x^8)^4(1-x)^{-5}=\\
[x^{21}]\sum_{i=0}^1 {1\choose i}(-x^9)^i\sum_{j=0}^4 {4\choose j}(-x^8)^j\sum_{k=0}^{\infty}{4+k\choose k}x^k=\\
\{(i,j,k)=(0,0,21),(0,1,11),(0,2,1),(1,0,12),(1,1,2)\}\\
{25\choose 21}-{4\choose 1}{15\choose 11}+{4\choose 2}{5\choose 1}-{16\choose 12}+{4\choose 1}{6\choose 2}=
5460.$$
WA answer.
