Q:

New Year Maths $2019$

$$\;\;\;\color{red}{\binom {20}{19}}\\
\color{orange}{+\binom {19}{a}
+{\binom  ab}}\\
\color{green}{+\binom bc
+\binom cd}\\
\color{blue}{+\binom de+
\binom ef}\\
\color{purple}{+\binom fg+\binom gh}\\
\color{magenta}{+\binom hk}\\[15pt]
\color{orange}=\\[10pt]
\color{red}{2019}\\[15pt]
\color{orange}{a,b,}
\color{green}{c,d,}
\color{blue}{e,f,}
\color{purple}{g,h,}
\color{magenta}k\color{orange}{=?}\\[20pt]$$
$$\color{red}{\text{Happy New Year!}}$$

Note - Please feel free to post any other interesting identities based on the same theme!

A:

$$\color{red}{\binom {20}{19}}
+\color{orange}{\binom {19}{16}
+{\binom  {16}{14}}}+ 
\color{green}{\binom {14}{12}
+\binom {12}{7}}+
\color{blue}{\binom {7}{6}+
\binom {6}{5}}+
\color{purple}{\binom {5}{3}+\binom {3}{1}}+
\color{magenta}{\binom {1}{0}}
=\color{red}{2019}$$

A:

$$\color{red}{\binom{20}{19}}+\color{orange}{\binom{19}{18}}+\color{orange}{\binom{18}{17}}+\color{green}{\binom{17}{16}}+\color{green}{\binom{16}{12}}+\color{blue}{\binom{12}{11}}+\color{blue}{\binom{11}{10}}+\color{purple}{\binom{10}{8}}+\color{purple}{\binom{8}{5}}+\color{magenta}{\binom{5}{0}}\color{orange}=\color{red}{2019}$$
$\color{red}{\text{Happy New Year!}}$

A:

A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> \cdots > k\geq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0

