How many poker hands can you choose with at least one face card present?

Question

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!

I would subtract $\binom{40}5$ too! – Angina Seng Aug 24 '17 at 11:56 — Angina Seng, Aug 24 '17 at 11:56
Either a typo or a big mistake! – Peter Aug 24 '17 at 12:25 — Peter, Aug 24 '17 at 12:25

score 1 · Accepted Answer · answered Aug 24 '17 at 13:02

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 $$\binom{52}{5}-\binom{40}{5}$$

How many poker hands can you choose with at least one face card present?

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