Unizor - Combinatorics - Advanced Problems 2.1.
Channel: Zor Shekhtman on UNIZOR Education
Date: 2014-06-16

Problem 1
A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Four of a Kind" combinations you can get?
The combination "Four of a Kind" is the one when you have four cards of the same rank (say, four 9's or four Kings) among five cards in your hand.
Answer: 624

Problem 2
A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Full House" combinations you can get?
The combination "Full House" is the one when you have three cards of one rank (say, three 8's or three Jacks) and two cards of another rank (say, two queens or two 5's) among five cards in your hand.
Answer: 3744

Problem 3
A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Straight" combinations you can get?
A "Straight" is a combination of five cards with sequential ranks of not of the same suit. For example, Queen, Jack, 10, 9, 8 of not of the same suit. If they are of the same suit, the combination is called "Straight Flush" and is considered to be a different combination.
There is one more rule for a "Straight". The ace of any suit can be the highest ranking card in a deck for a "Straight" combination Ace, King, Queen, Jack, 10 or the lowest ranking card in a deck with the rank of 1 in a combination 5, 4, 3, 2, A.
Answer: 10,200

Problem 4
A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Flush" combinations you can get?
A "Flush" is a combination of five cards of the same suit with ranks not sequentially following each other (then it would be "Straight Flush" combination, which is different from "Flush").
Answer: 5,108

Problem 5
A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Three of a Kind" combinations you can get?
The combination "Three of a Kind" is the one when you have three cards of the same rank (say, three 9's or three Kings) among five cards in your hand, and the other two cards of not of this rank (otherwise, you would have "Four of a Kind" combination) and not of the same rank among themselves (otherwise, you would have "Full House" combination).
Answer: 54,912