Description
Poker Hands
If you have played poker, you probably know some or all the hands below [1]. You can choose 5 cards from 52 in ways. But how many of them would be a Royal Flush or a Four-of-a-Kind?
Let’s try to calculate the numbers for all the following hands.
- Royal Flush: All five cards are of the same suit and are of the sequence 10 J Q K
- Straight Flush: All five cards are of the same suit and are sequential in rank.
- Four-of-a-Kind: Four cards are all of the same rank.
- Flush: All five cards are of the same suit but not all sequential in rank.
- Straight: All five cards are sequential in rank but are not all of the same suit.
- Three-of-a-Kind: Three cards are all of the same rank and the other two are each of different ranks from the first three and each other.
- Two Pair: Two pairs of two cards of the same rank (the ranks of each pair are different in rank, obviously, to avoid a Four-of-a-Kind)
- One Pair: Only two cards of the five are of the same rank with the other three cards all having different ranks from each other and from that of the pair.
- Full House: A hand consisting of one pair and a three-of-a-kind of a different rank than the pair.
1.2 Some other problems
- Six friends want to play enough games of chess and every one wants to play everyone else. How many games will they have to play?
- There are five flavours of ice cream: banana, chocolate, lemon, strawberry and vanilla. We can have three scoops. How many variations will there be? [2]
- For 𝑥 a real number and 𝑛 a positive integer, show that
- Determine the number of integer solutions of 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 32, where each 𝑥𝑖 ≥
- Let 𝐴 be a subset of {1,2,3, … ,25} where |𝐴| = 9. For any subset 𝐵 of 𝐴 let 𝑠𝐵 denote the sum of the elements in 𝐵. Prove that there are distinct subsets 𝐶, 𝐷 of 𝐴 such that |𝐶| = |𝐷| = 5 and 𝑠𝐶 = 𝑠𝐷.