Number of 5-card Charlie hands in blackjack

Question

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?

Answer 1

Here is an answer by brute force enumeration of all 5-card hands, using an R program: the number of 5-card Charlie hands is 139,972.

deck <- c(rep(1:9, 4), rep(10, 16))
acceptable <- function(x) {sum(x) <= 21}
sum(combn(deck, 5, acceptable))