Answer must be [math]4851[/math]. This is because :

We can divide the set of spades into two segments namely “ Face cards and Non-face cards”.

CASE 1:

Now if the spade card is also a face card then :

Total choices for [math]5[/math] face cards in the hand [math] \displaystyle \binom {12}{5}.[/math]

Now, [math]\displaystyle \binom {9}{5}[/math] would be non-spade face cards choices and [math]\displaystyle \binom {8}{5}[/math] would be no King choices and [math]\displaystyle \binom {6}{5}[/math] would have neither.

By Inclusion-Exclusion Principle required choices would be : [math] \displaystyle \binom {12}{5} - \binom {9}{5} - \binom {8}{5} + \binom {6}{5}.[/math]

But here we have counted a set where there would be one-spade king. Thus it would be one card instead of two, if and only if there is no other spade or king. Thus we need to subtract those cases which would be: [math]\displaystyle \binom {6}{4}.[/math]

CASE 2:

Now if the spade card is not a face card then :

There are [math]10[/math] choices for spade card, and for the remaining [math]4[/math] face cards in the hand, there would be [math]\displaystyle \binom {12}{4}[/math] total choices, with [math]\displaystyle \binom {8}{4}[/math] no king card choices.

In this case, total number of choices are : [math] \displaystyle 10 ( \binom {12}{4} - \binom {8}{4} ).[/math]

So the answer is : [math] \displaystyle \binom {12}{5} - \binom {9}{5} - \binom {8}{5} + \binom {6}{5} - \binom {6}{4} + 10 ( \binom {12}{4} - \binom {8}{4} ) = 4851 .[/math]