# Permutations examples

If you are not familiar with the n! A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once? For the first letter, there are 5 possible choices. After that letter is chosen, there are 4 possible choices.

Finally, there are 3 possible choices. In how many ways can a president, a treasurer and a secretary be chosen from among 7 candidates? For the first position, there are 7 possible choices. After that candidate is chosen, there are 6 possible choices. Finally, there are 5 possible choices.

### Permutations P(n,r)

There are possible ways to choose a president, a treasurer and a secretary be chosen from among 7 candidates. A zip code contains 5 digits. How many different zip codes can be made with the digits 0—9 if no digit is used more than once and the first digit is not 0? For the first position, there are 9 possible choices since 0 is not allowed. After that number is chosen, there are 9 possible choices since 0 is now allowed.

Then, there are 8 possible choices, 7 possible choices and 6 possible choices. For the next 4 positions, we are selecting from 9 digits.

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Related Topics: More Lessons on Probability In this lesson, we will learn the permutation formula for the number of permutations of n things taken r at a time.

We will also learn how to solve permutation word problems with repeated symbols and permutation word problems with restrictions or special conditions.

What is Permutation? A permutation is an arrangement, or listing, of objects in which the order is important. In previous lessonswe looked at examples of the number of permutations of n things taken n at a time. Permutation is used when we are counting without replacement and the order matters. If the order does not matter then we can use combinations. The following diagrams give the formulas for Permutation, Combination, and Permutation with Repeated Symbols.

Scroll down the page with more examples and step by step solutions. What is the Permutation Formula? In general P nr means that the number of permutations of n things taken r at a time. We can either use reasoning to solve these types of permutation problems or we can use the permutation formula. The formula for permutation is If you are not familiar with the n! You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics.In a group of 6 boys and 4 girls, four children are to be selected such that at least one boy should be there.

Hence we have 4 options as given below We can select 4 boys From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be done? From a group of 7 men and 6 women, five persons are to be selected with at least 3 men. Hence we have the following 3 options. We can select 5 men It has the vowels 'O','I','A' in it and these 3 vowels should always come together. Hence these three vowels can be grouped and considered as a single letter.

Hence we can assume total letters as 5 and all these letters are different. Hence these 5 vowels can be grouped and considered as a single letter. Hence we can assume total letters as 7. But in these 7 letters, 'R' occurs 2 times and rest of the letters are different. We use cookies to personalise content and ads, to provide social media features and to analyse our traffic.

We also share information about your use of our site with our social media, advertising and analytics partners. More information Agree. Custom Search. Solved Examples Set 1 - Permutation and Combination 1. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? Answer: Option B Explanation: In a group of 6 boys and 4 girls, four children are to be selected such that at least one boy should be there.

Answer: Option C Explanation: From a group of 7 men and 6 women, five persons are to be selected with at least 3 men.

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Jay You are looking for integer partitions. Refer Counting Integral Solutions. Kushal Vyas There are three proof: A Pattern 4!

B Practical sums: No. But other24 alphabets are not there so they are 0 in no. For them it will be 0! Ways If there are 2 items we can arrange in 2! Ways If there are 1 items we can arrange in1! Ways If there are 0 items we can arrange in 1 way Manu Hai,Bro what u said is right and simply we can learn by C programming for 0!

Don't think bad its just an example for knowing the factorial case.And it is. With permutations, every little detail matters. Combinations, on the other hand, are pretty easy going. A joke: A "combination lock" should really be called a "permutation lock". The order you put the numbers in matters. A true "combination lock" would accept both and as correct. How many ways can we award a 1st, 2nd and 3rd place prize among eight contestants? We had to order 3 people out of 8. To do this, we started with all options 8 then took them away one at a time 7, then 6 until we ran out of medals.

Unfortunately, that does too much! And why did we use the number 5? Because it was left over after we picked 3 medals from 8. So, a better way to write this would be:. If we have n items total and want to pick k in a certain order, we get:. And this is the fancy permutation formula: You have n items and want to find the number of ways k items can be ordered:. Combinations are easy going. You can mix it up and it looks the same. In fact, I can only afford empty tin cans. Well, we have 3 choices for the first person, 2 for the second, and only 1 for the last. Indeed I did. So, if we have 3 tin cans to give away, there are 3! If we want to figure out how many combinations we have, we just create all the permutations and divide by all the redundancies. Writing this out, we get our combination formulaor the number of ways to combine k items from a set of n:.

Combination: Picking a team of 3 people from a group of Combinations sound simpler than permutations, and they are. You have fewer combinations than permutations. BetterExplained helps k monthly readers with friendly, insightful math lessons more. Easy Permutations and Combinations. About The Site BetterExplained helps k monthly readers with friendly, insightful math lessons more.The same rule applies while solving any problem in Permutations. Therefore, the number of ways in which the 3 letters can be arranged, taken all a time, is 3!

The different ways in which the 3 letters, taken 2 at a time, can be arranged is 3! When a letter occurs more than once in a word, we divide the factorial of the number of all letters in the word by the number of occurrences of each letter. Solution :. In order to find the number of permutations that can be formed where the two vowels U and E come together.

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In these cases, we group the letters that should come together and consider that group as one letter. After 3 vowels take 3 places, no.

It does not matter whether we select A after B or B after A. The order of selection is not important in combinations. To find the number of combinations possible from a given group of items n, taken r at a time, the formula, denoted by n C r is. For example, verifying the above example, the different selections possible from the alphabets A, B, C, taken two at a time are.

Problem 1: In how many ways can a committee of 1 man and 3 women can be formed from a group of 3 men and 4 women? Problem 2: Among a set of 5 black balls and 3 red balls, how many selections of 5 balls can be made such that at least 3 of them are black balls. Selecting at least 3 black balls from a set of 5 black balls in a total selection of 5 balls can be. Therefore, our solution expression looks like this.

Problem 3: How many 4 digit numbers that are divisible by 10 can be formed from the numbers 3, 5, 7, 8, 9, 0 such that no number repeats?

After 0 is placed in the units place, the tens place can be filled with any of the other 5 digits. After filling the tens place, we are left with 4 digits. In how many ways can a selection of 3 men and 2 women can be made from a group of 5 men and 5 women? How many of them begin with C?

How many of them begin with T? In how many of them consonants will occur together? Correct me if its worng! How the answer of 3 is 2! As per my understanding it should be 10!To have no repeated digits, all four digits would have to be different, which is selecting without replacement.

The probability of no repeated digits is the number of 4 digit PINs with no repeated digits divided by the total number of 4 digit PINs. This probability is. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. Compute the probability that you win the second prize if you purchase a single lottery ticket. In order to win the second prize, five of the six numbers on the ticket must match five of the six winning numbers; in other words, we must have chosen five of the six winning numbers and one of the 42 losing numbers.

So the probability of winning the second prize is. A multiple-choice question on an economics quiz contains 10 questions with five possible answers each. Compute the probability of randomly guessing the answers and getting exactly 9 questions correct. In many card games such as poker the order in which the cards are drawn is not important since the player may rearrange the cards in his hand any way he chooses ; in the problems that follow, we will assume that this is the case unless otherwise stated.

Thus we use combinations to compute the possible number of 5-card hands, 52 C 5. This number will go in the denominator of our probability formula, since it is the number of possible outcomes. For the numerator, we need the number of ways to draw one Ace and four other cards none of them Aces from the deck. Since there are four Aces and we want exactly one of them, there will be 4 C 1 ways to select one Ace; since there are 48 non-Aces and we want 4 of them, there will be 48 C 4 ways to select the four non-Aces.

The solution is similar to the previous example, except now we are choosing 2 Aces out of 4 and 3 non-Aces out of 48; the denominator remains the same:.

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It is useful to note that these card problems are remarkably similar to the lottery problems discussed earlier.

Compute the probability of randomly drawing five cards from a deck of cards and getting three Aces and two Kings. Suppose you have a room full of 30 people.

What is the probability that there is at least one shared birthday? Take a guess at the answer to the above problem. Suppose three people are in a room.Permutations and combinationsthe various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.

By considering the ratio of the number of desired subsets to the number of all possible subsets for many games of chance in the 17th century, the French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory.

## Permutations and combinations

The concepts of and differences between permutations and combinations can be illustrated by examination of all the different ways in which a pair of objects can be selected from five distinguishable objects—such as the letters A, B, C, D, and E.

If both the letters selected and the order of selection are considered, then the following 20 outcomes are possible:.

Each of these 20 different possible selections is called a permutation. The expression n! For example, using this formula, the number of permutations of five objects taken two at a time is. Thus, for 5 objects there are 5!

For combinations, k objects are selected from a set of n objects to produce subsets without ordering. This is the same as the nk binomial coefficient see binomial theorem. For example, the number of combinations of five objects taken two at a time is. The formulas for n P k and n C k are called counting formulas since they can be used to count the number of possible permutations or combinations in a given situation without having to list them all.

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Thank you for your feedback. Permutations and combinations mathematics. Read More on This Topic. The number of permutations is…. Get exclusive access to content from our First Edition with your subscription. Subscribe today. Learn More in these related Britannica articles:. A permutation of a set, say the elements aband cis any re-ordering of the elements, and it is usually denoted as follows:This particular permutation takes a to cb to aand c to b.

For three elements, as here, there…. History at your fingertips. Sign up here to see what happened On This Dayevery day in your inbox! Email address. By signing up, you agree to our Privacy Notice. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. More About.In English we use the word "combination" loosely, without thinking if the order of things is important.

In other words:. Now we do care about the order. It has to be exactly More generally: choosing r of something that has n different types, the permutations are:. In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.

Example: in the lock above, there are 10 numbers to choose from 0,1,2,3,4,5,6,7,8,9 and we choose 3 of them:. So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, 12, 11, And the total permutations are:. In other words, there are 3, different ways that 3 pool balls could be arranged out of 16 balls.

But how do we write that mathematically?

### Combinations and Permutations

Answer: we use the " factorial function ". The factorial function symbol:! But when we want to select just 3 we don't want to multiply after How do we do that? There is a neat trick: we divide by 13! That was neat.

This is how lotteries work.

Permutations and Combinations - word problems 128-1.11

The numbers are drawn one at a time, and if we have the lucky numbers no matter what order we win! Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order. In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:. Another example: 4 things can be placed in 4! So we adjust our permutations formula to reduce it by how many ways the objects could be in order because we aren't interested in their order any more :.

In other words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations. We can also use Pascal's Triangle to find the values. Go down to row "n" the top row is 0and then along "r" places and the value there is our answer. Here is an extract showing row Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla. Order does not matter, and we can repeat! Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out. Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate! So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.

We can write this down as arrow means movecircle means scoop.