You know that getting 2 cards to 21 means having (10,11). Now, to get 3 cards to 21, instead of starting from scratch, why not start from the first two (10,11)? Choose one of the cards to “break apart”, for example 10 => 4,6. So you get ((4,6), 11). I think one approach would be to figure out how to recursively break apart the pieces (You can again break 6 into (1,5)).

Off the top of my head, that’s my first instinct for approaching it .

Great website you’ve got here, as an engineer my maths is pretty good, but statistics is one area of maths I’ve never studied. Your explanations are better than most I’ve come across on mathematics websites and textbooks.

I have a question that I wonder if there is an easy answer to:

‘How many permutations of 4 cards (value 1 to 11) sum to a value of 21?’

I can do this the long way, but I’m looking for a method or formula to speed up the process.

For example, for 2 cards there are 2 permutations that total 21:

10 , 11
11 , 10

for 3 cards there are x permutations that total 21:

1 , 9 , 11
1 , 10, 10
1, 11, 9
2, 8, 11
2, 9, 10
etc.

By the time I consider 4 cards however, this method is too long-winded and cumbersome to be of any use.

Do you know of a quicker way to arrive at the answer than simply listing all of the permutations and counting them?

Thanks.