$\left(\begin{array}{c}10\\ 3\end{array}\right)=120$

$2^{10}=1024$

$\left(\begin{array}{c}10\\ 8\end{array}\right)+\left(\begin{array}{c}10\\ 9\end{array}\right)+\left(\begin{array}{c}10\\ 10\end{array}\right)=56$

$1024-\left[\left(\begin{array}{c}10\\ 9\end{array}\right)+\left(\begin{array}{c}10\\ 10\end{array}\right)\right]=1013$

Draw grid.

$\left(\begin{array}{c}5\\ 2\end{array}\right)=10$

$32$

$\left(\begin{array}{c}14\\ 4\end{array}\right)=1001$

$\left(\begin{array}{c}10\\ 2\end{array}\right)=45$

1 - 0, 2 - 0, 2 - 1, 3 - 1, 4 - 1, 4 - 2, 4 - 3, 5 - 3, 6 - 3, 6 - 4.

$210$

$40$

$2^5=32$

$30$

There are exactly $\left(\begin{array}{c}5\\ 2\end{array}\right)=10$ possibilities:
1: - - — — —
2: - — - — —
3: - — — - —
4: - — — — -
5: — - - — —
6: — - — - —
7: — - — — -
8: — — - - —
9: — — - — -
0: — — — - -