Number of 5-card Charlie hands in blackjack A five-card Charlie in blackjack is when you have a total of 5 cards and you do not exceed a point total of 21. How many such hands are there? Of course, the natural next question concerns six-card Charlies, etc. 
It seems like one way of determining the answer might be to determine the total number of 5-card hands and then subtract out the number of hands that exceed 21, but I am at a loss as to how to do this effectively. Is there some use of the inclusion-exclusion principle at work here? The condition that the cards do not exceed 21 is the difficulty I am having a hard time addressing. Any ideas?

One way to classify all blackjack hands is with a generating function:
$$
   \left((x y+1) \left(x^2 y+1\right) \left(x^3 y+1\right) \left(x^4 y+1\right) \left(x^5 y+1\right) \left(x^6 y+1\right) \left(x^7 y+1\right) \left(x^8 y+1\right) \\ \left(x^9 y+1\right) \left(x^{10} y+1\right)^4\right)^4
$$
Here, the coefficient of $x^a y^b$ gives us the number of hands that add up to $a$ and contain $b$ cards. (Assuming that all aces are worth $1$.)
So we can use the following Mathematica code to extract the information we want.
First, define the generating function above:
gf = ((1 + x y) (1 + x^2 y) (1 + x^3 y) (1 + x^4 y) (1 + x^5 y) (1 + x^6 y) (1 + x^7 y) (1 + x^8 y) (1 + x^9 y) (1 + x^10 y)^4)^4;

Next, take the coefficient of $y^5$ to only look at $5$-card hands:
poly = Coefficient[gf, y^5];

Extract a list of the coefficients of $x$: this has the number of hands with each possible value.
coeffs = CoefficientList[poly, x];

Finally, add up the first $22$ coefficient (the values $0$ through $21$).
Total[coeffs[[1 ;; 22]]]

The second line can be modified in the obvious way to get different hand sizes.
