How many poker hands can you choose with at least one face card present (Jack, Queen or King)? Answer I found was:
$${\binom {52}{5}} – {\binom {48}{5}}$$
Stumbled across this one and I don't really understand how I should think about it. Why is ${\tbinom {48}{5}}$ the way to cut out all combinations that do not furfill the criteria? The way I'm thinking is that you'd want to create a hand with purely "number" cards (i.e. from 1-10) therefore subtracting ${\tbinom {40}{5}}$ instead.
What am I missing in my way of thinking?
Thanks!
Best Answer
My hunch is that answer you found is wrong and your intuition is right. The total number of poker hands with a regular 52 card pack is $\tbinom{52}{5}$. The total number of hands without figure cards is $\tbinom{40}{5}$ as there are 12 cards with figures (4 x J, Q, K). So the number of hands with at least one figure is $$\binom{52}{5}-\binom{40}{5}$$