Hello, amyl95!

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(1) How many different poker hands can be dealt to a player?

You are correct!

There are: .$\displaystyle {52\choose5} \:=\:\frac{52!}{5!\,47!} \:=\:2,\!598,\!960$ possible hands.

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(2) How many hands include the Ace of Spades?

We want the $\displaystyle A\spadesuit$ and four of the 51 other cards.

. . $\displaystyle 1\cdot{51\choose4} \:=\:249,\!900$

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(3) How many hands do not contain the Ace of Spades?

We want 5 of the other 51 cards.

. . $\displaystyle {51\choose5} \:=\:2,\!349,\!060$

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(4) How many hands include all 4 Aces?

There is one way to have all four Aces.

And we want one of the other 48 cards.

. . $\displaystyle 1\cdot{48\choose1} \:=\:48$

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(5) How many hands contain exactly 3 Aces?

There are $\displaystyle {4\choose3}$ choices for the three Aces.

And there are $\displaystyle {48\choose2}$ ways to choose the other two cards.

. . $\displaystyle {4\choose3}{48\choose2} \:=\:4\cdot 1128 \:=\:4512$