A standard deck of cards has $52$ cards divided into $4$ suits, each of which has $13$ cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usually by a process called 'shuffling'). In how many ways can we pick five different cards? (Order matters, thus ace of spades followed by jack of diamonds is different than jack of diamonds followed by ace of spades.)

Question

A standard deck of cards has $52$ cards divided into $4$ suits, each of which has $13$ cards.  Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black.  The cards in the deck are placed in random order (usually by a process called 'shuffling').  In how many ways can we pick five different cards? (Order matters, thus ace of spades followed by jack of diamonds is different than jack of diamonds followed by ace of spades.)

GPT 3.5 · Answer

There are 52 choices for the first card, 51 choices for the second card, and continue this way until we reach the 5$^{	ext{th}}$ card, which has 48 choices. That gives a total of $52	imes51	imes50	imes49	imes48=165,\!765,\!600$ for choosing 5 cards. However, the order in which we pick the cards matters, so we must multiply by $5!$ (the number of ways to arrange the 5 cards). This gives us a final answer of $165,\!765,\!600	imes5!=\boxed{40,\!270,\!144,\!000}$.