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.
1. Royal Flush: All five cards are of the same suit and are of the sequence 10 J Q K A.
2. Straight Flush: All five cards are of the same suit and are sequential in rank.
3. Four-of-a-Kind: Four cards are all of the same rank.
4. Flush: All five cards are of the same suit but not all sequential in rank.
5. Straight: All five cards are sequential in rank but are not all of the same suit.
6. 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.
7. 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)
8. 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.
9. 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 1. 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?
2. 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]
3. For 𝑥 a real number and 𝑛 a positive integer, show that
4. Determine the number of integer solutions of 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 32, where each 𝑥𝑖 ≥
0.
5. 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.
