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.)

Answer 1

There are 52 choices for the first card, 51 choices for the second card, and continue this way until we reach the 5$^{\text{th}}$ card, which has 48 choices. That gives a total of $52\times51\times50\times49\times48=165,\!765,\!600$ 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 $165,\!765,\!600\times5!=\boxed{40,\!270,\!144,\!000}$.

Answer 2

To find the number of ways to pick five different cards from a standard deck of 52 cards, we can consider the problem in two steps:

1. Selecting the first card: Since order matters, we have 52 choices for the first card.

2. Selecting the remaining four cards: After the first card is selected, there are 51 cards left in the deck. For each subsequent card, the number of choices reduces by one. Therefore, we have 51 choices for the second card, 50 choices for the third card, 49 choices for the fourth card, and 48 choices for the fifth card.

To determine the total number of ways, we multiply the choices at each step:

$52 \times 51 \times 50 \times 49 \times 48 = 311,875,200$

Hence, there are 311,875,200 ways to select five different cards from a standard deck.

Answer 3

To solve this problem, we need to determine the number of ways we can select five different cards from a standard deck of $52$ cards, where order matters.

We can break down the problem into two steps:

1. Selecting the first card: Since order matters, we can select any of the $52$ cards from the deck as the first card.

2. Selecting the remaining four cards: After selecting the first card, we now have $51$ remaining cards to choose from. Since order matters, we can select any of the $51$ remaining cards as the second card. Similarly, after selecting the second card, there are $50$ remaining cards to choose from for the third card, $49$ for the fourth card, and $48$ for the fifth card.

To find the total number of ways to select five different cards, we multiply the number of choices at each step:

Number of ways = $52 \cdot 51 \cdot 50 \cdot 49 \cdot 48$

To evaluate this expression, we can use a calculator or simplify it further if desired.