Example: 9 card hands | Probability and combinatorics | Precalculus | Khan Academy

TL;DR

There are 94,143,280 possible 9-card hands in the card game described.

Transcript

A card game using 36 unique cards, four suits, diamonds, hearts, clubs and spades-- this should be spades, not spaces-- with cards numbered from 1 to 9 in each suit. A hand is chosen. A hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are... Read More

Key Insights

• 🗂️ The card game includes 36 unique cards divided into 4 suits, with 9 cards in each suit.
• #️⃣ The number of possible 9-card hands is calculated by multiplying the number of choices for each slot.
• #️⃣ When calculating possible combinations, the formula n choose k is used, where n is the total number of options and k is the number of selections.
• 🗂️ The order of the cards in a hand does not matter, so the formula is adjusted to account for this by dividing by the number of ways the cards can be rearranged.

Questions & Answers

Q: How many unique cards are there in the card game?

There are 36 unique cards, with 4 suits, each containing 9 numbered cards.

Q: How many cards are there in a hand?

A hand consists of 9 cards.

Q: Do players have the freedom to arrange the cards in any order?

Yes, the player can sort the cards in any way they choose.

Q: Why is it necessary to divide the total number of possible hands by the ways the cards can be rearranged?

By dividing, we prevent overcounting the different ways the same cards can be arranged and ensure we only count unique hands.

Q: What is the formula used to calculate the total number of possible hands?

The formula is n choose k, which is equal to n factorial divided by (n - k) factorial multiplied by k factorial.

Summary & Key Takeaways

• The card game includes 36 unique cards divided into 4 suits, each suit containing 9 numbered cards.

• A hand consists of 9 cards, with the player being able to arrange them in any order.

• The total number of possible 9-card hands is calculated by multiplying the number of choices for each slot (36, 35, 34, etc.) and then dividing by the number of ways the cards can be ordered (9 factorial).