To solve this problem, we need to calculate the number of ways to choose at least 3 spades from a standard deck of 52 cards.

Step 1: Breakdown of the deck

• A standard deck has 52 cards.
• There are 13 spades in the deck.
• There are 39 non-spade cards in the deck.

Step 2: Total number of ways to choose 5 cards from 52

The total number of ways to select 5 cards from 52 is given by the combination:

$\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960$

Step 3: Cases for choosing at least 3 spades

We need to consider the following cases:

• 3 spades and 2 non-spades
• 4 spades and 1 non-spade
• 5 spades

Case 1: 3 spades and 2 non-spades

We can choose 3 spades from the 13 spades, and 2 non-spades from the 39 non-spades:

$\binom{13}{3} \times \binom{39}{2} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} \times \frac{39 \times 38}{2 \times 1} = 286 \times 741 = 211,926$

Case 2: 4 spades and 1 non-spade

We can choose 4 spades from the 13 spades, and 1 non-spade from the 39 non-spades:

$\binom{13}{4} \times \binom{39}{1} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} \times 39 = 715 \times 39 = 27,885$

Case 3: 5 spades

We can choose all 5 cards from the 13 spades:

$\binom{13}{5} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1,287$

Step 4: Total number of ways to choose at least 3 spades

Now, we add up all the cases:

$211,926 + 27,885 + 1,287 = 241,098$

Conclusion:

There are 241,098 ways to choose at least 3 spades from a deck of 52 cards.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How many ways are there to choose exactly 4 spades in a poker hand?
2. How many ways are there to choose a hand with no spades?
3. What is the probability of drawing exactly 3 spades from a deck of 52 cards?
4. How many ways can you choose 5 cards from the hearts suit only?
5. How would the solution change if there were 54 cards in the deck, including jokers?

Tip: The combination formula $\binom{n}{r}$ is crucial for counting selections without regard to order, and is extensively used in probability and combinatorics.