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Permutations
Permutations and Combinations
Counting Methods
Factorial Lessons
Probability

What Is Combination In Math?

An arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C. In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.

What Is The Combination Formula?

The number of different possible poker hands is found by counting the number of ways that 5 cards can be selected from 52 cards, where the order is not important. It is a combination, so we use `Cr^n`. Poker Mathematics Poker is a game of skill and using the ability to read situations and opponents to give you the advantage in each hand you play. It is also a game of mathematics, where you should be able to calculate the odds of either you or your opponent winning the hand in any situation.

The number of combinations of n things taken r at a time is written as C(n, r).

The following diagram shows the formula for combination. Scroll down the page for more examples and solutions on how to use the combination formula.

In a standard deck of cards, there are 4 possible suits (clubs, diamonds, hearts, spades), and 13 possible values (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace). Let A,J,Q,K represent Ace, Jack, Queen and King, respectively. Every card has a suit and value, and every combination is possible. Hence a standard deck contains 134 = 52 cards.

If you are not familiar with the n! (n factorial notation) then have a look the factorial lesson

How To Use The Combination Formula To Solve Word Problems?

Example:
In how many ways can a coach choose three swimmers from among five swimmers?

Solution:
There are 5 swimmers to be taken 3 at a time.
Using the formula:

The coach can choose the swimmers in 10 ways.

Example:
Six friends want to play enough games of chess to be sure every one plays everyone else. How many games will they have to play?

Solution:
There are 6 players to be taken 2 at a time.
Using the formula:

They will need to play 15 games.

Example:
In a lottery, each ticket has 5 one-digit numbers 0-9 on it.
a) You win if your ticket has the digits in any order. What are your changes of winning?
b) You would win only if your ticket has the digits in the required order. What are your chances of winning?

Solution:
There are 10 digits to be taken 5 at a time.

a) Using the formula:

The chances of winning are 1 out of 252.

b) Since the order matters, we should use permutation instead of combination.
P(10, 5) = 10 x 9 x 8 x 7 x 6 = 30240

The chances of winning are 1 out of 30240.

How To Evaluate Combinations As Well As Solve Counting Problems Using Combinations?

A combination is a grouping or subset of items. For a combination, the order does not matter.

How many committees of 3 can be formed from a group of 4 students?
This is a combination and can be written as C(4,3) or ₄C₃ or (left( {begin{array}{*{20}{c}}43end{array}} right)).

Examples:

1. The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be on the field at any one time?
2. A student need 8 more classes to complete her degree. If she met the prerequisites for all the courses, how many ways can she take 4 classes next semester?
3. There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 man and 2 women. How many different groups can be formed from the office?

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How To Solve Combination Problems That Involve Selecting Groups Based On Conditional Criteria?

Example:A bucket contains the following marbles: 4 red, 3 blue, 4 green, and 3 yellow making 14 total marbles. Each marble is labeled with a number so they can be distinguished.

1. How many sets/groups of 4 marbles are possible?
2. How many sets/groups of 4 are there such that each one is a different color?
3. How many sets of 4 are there in which at least 2 are red?
4. How many sets of 4 are there in which none are red, but at least one is green?

How To Solve Word Problems Involving Permutations And Combinations?

Examples:

1. A museum has 7 paintings by Picasso and wants to arrange 3 of them on the same wall. How many ways are there to do this?
2. How many ways can you arrange the letters in the word LOLLIPOP?
3. A person playing poker is dealt 5 cards. How many different hands could the player have been dealt?

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Brian Alspach

13 January 2000

Abstract:

The types of 5-card poker hands are

• straight flush
• 4-of-a-kind
• full house
• flush
• straight
• 3-of-a-kind
• two pairs
• a pair
• high card

Most poker games are based on 5-card poker hands so the ranking ofthese hands is crucial. There can be some interesting situationsarising when the game involves choosing 5 cards from 6 or more cards,but in this case we are counting 5-card hands based on holding only5 cards. The total number of 5-card poker hands is.

A straight flush is completely determined once the smallest card in thestraight flush is known. There are 40 cards eligible to be the smallestcard in a straight flush. Hence, there are 40 straight flushes.

In forming a 4-of-a-kind hand, there are 13 choices for the rank ofthe quads, 1 choice for the 4 cards of the given rank, and 48 choicesfor the remaining card. This implies there are 4-of-a-kind hands.

There are 13 choices for the rank of the triple and 12 choices for therank of the pair in a full house. There are 4 ways of choosing thetriple of a given rank and 6 ways to choose the pair of the other rank.This produces full houses.

To count the number of flushes, we obtain choicesfor 5 cards in the same suit. Of these, 10 are straight flushes whoseremoval leaves 1,277 flushes of a given suit. Multiplying by 4 produces5,108 flushes.

The ranks of the cards in a straight have the form x,x+1,x+2,x+3,x+4,where x can be any of 10 ranks. There are then 4 choices for each card ofthe given ranks. This yields total choices. However,this count includes the straight flushes. Removing the 40 straightflushes leaves us with 10,200 straights.

In forming a 3-of-a-kind hand, there are 13 choices for the rank of thetriple, and there are choices for the ranks of theother 2 cards. There are 4 choices for the triple of the given rank andthere are 4 choices for each of the cards of the remaining 2 ranks.Altogether, we have 3-of-a-kind hands.

Poker Combinations Math Test

Next we consider two pairs hands. There are choices for the two ranks of the pairs. There are 6 choices for eachof the pairs, and there are 44 choices for the remaining card. Thisproduces hands of two pairs.

Now we count the number of hands with a pair. There are 13 choices forthe rank of the pair, and 6 choices for a pair of the chosen rank. Thereare choices for the ranks of the other 3 cardsand 4 choices for each of these 3 cards. We have hands with a pair.

Poker Combinations Math Cheat

We could determine the number of high card hands by removing the handswhich have already been counted in one of the previous categories.Instead, let us count them independently and see if the numbers sumto 2,598,960 which will serve as a check on our arithmetic.

Poker Combinations Math Game

A high card hand has 5 distinct ranks, but does not allow ranks of theform x,x+1,x+2,x+3,x+4 as that would constitute a straight. Thus, thereare possible sets of ranks from which we remove the10 sets of the form .This leaves 1,277 sets of ranks.For a given set of ranks, there are 4 choices for each cardexcept we cannot choose all in the same suit. Hence, there are1277(4⁵-4) = 1,302,540 high card hands.

If we sum the preceding numbers, we obtain 2,598,960 and we can be confidentthe numbers are correct.

Here is a table summarizing the number of 5-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 5 cards.