Walker14
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From a standard 52-card deck, how many 7-card hands have exactly 5 spades and 2 hearts?
Walker14 said:From a standard 52-card deck, how many 7-card hands have exactly 5 spades and 2 hearts?
Jameson said:Hi Walker14,
Welcome to MHB! :)
What have you tried so far? With problems like this it is very useful to use combinations. You might have seen it written like this:
$${n \choose k}$$
Prove It said:Hi Jameson, Walker14 is a student of mine. This problem came up during tutoring and I had a brain fart - unfortunately we couldn't figure out where to start, as there was so many variables to deal with - not only figuring out how many ways there are to get spades, but also how to arrange 5 of them, and then the same with hearts...
Choose 5 from the available 13 spades: {13\choose5} = 1287 ways.From a standard 52-card deck, how many 7-card hands
have exactly 5 spades and 2 hearts?
Opalg said:This had me in all sorts of trouble for a while, but I ended up realising that it is as simple as Jameson indicates. The number of arrangements is ${13 \choose 5}{13 \choose 2} = 100\,386$. In terms of probabilities, the chances of drawing such a hand from a 52-card pack are $$\frac{{13 \choose 5}{13 \choose 2}}{52\choose7} \approx 0.00075035565.$$
Jameson said:These kinds of problems are either this easy or incredibly tedious. I pretty much always feel doubt with which ever way I go and I know that if it's not just multiplying some combinations together then I better set aside an afternoon to and grab a lot of paper.