...PERMUTATION...
What is permutation?? People always get confused "permutation" and "combination" which one's which??
Introduction of permutation
Permutation sound complicated, doesn't?? and it is "Permutation is every little detail matters". Permutation actually an arrangement of all or part of a set of objects, with regard to the order of the arrangement.
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. These differ from combinations, which are selections of some members of a set where order is disregarded.
Meaning of permutation
A permutation is an arrangement, or listing, of objects in which the order is important. In the previous lessons, we 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.
Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.
In general P(n, r) 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
Example 1:
A zip code contains 5 digits. How many different zip code can be made with the digit 0 - 9 if no digit is used more than once and the first digit is not 0?
Solution:
Using reasoning:
for the first position, there are 9 possible choice (since 0 is not allowed). After that number is chosen, there are 9 possible choices (since 0 is not allowed). Then there are 8 possible choices, 7 possible choices and 6 possible choices.
9 x 9 x 8 x 7 x 6 = 27,216
Using permutation formula:
We can't include the first digit in the formula because 0 is not allowed. For the first position, there are 9 possible choices (since 0 is not allowed). For the next 4 possitions, we are selecting from 9 digits.
9 x P(9,4) = 9 x 9! / (9 - 4)! = 9 x 9! / 5! = 9 x 9 x 8 x 7 x 6 x5! / 5! = 27,216
Example 2:
if five digit 1,2, 3, 4, 5 are being given and a three digit code has to be made from it if the repetition of digits is allowed then how many such code can be formed.
Solution:
As repetition is allowed, we have five options for each digit of the code. Hence the required number of ways code can be formed is 5 x 5 x 5 = 125.
Example 3:
if three alphabets are to be chosen from A, B, C, D and E such that repetition is not allowed then in how many ways it can be dome?
Solution:
The number of ways three alphabets can be chosen from five will be,
3 / 5p = 5! / (5-3)! = 5 x 4 x 3 x 2 x1 / 2 x 1 = 60
Hence, there are 60 possible ways.
Permutations Word Problems
Example 1:
In how many ways can the letters of the word APPLE can be rearranged?
Solution:
Total number of alphabets in APPLE = 5.
Number of repeated alphabets = 2
Number of ways APPLE can be rearranged =
The word APPLE can be rearranged in 60 ways.
Example 2:
10 students have appeared in a test in which the top three will get a prize. How many possible ways are there to get the prize winners?
Solution:
We need to choose and arrange 3 persons out of 10. Hence, the number of possible ways will be
Example 3:
Ellie want to change her password which is ELLIE9 but with same letters and number. In how many ways she can do that?
Solution:
Total number of letters = 6.
Repeated letters = 2 Ls and 2 Es.
Number of times ELLIE9 can be rearranged =
But the password need to be changed. So, the number of ways new password can be made =
This is the video for permutation. let's learn from the basic..
...COMBINATION...
Introduction:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination. Combination doesn't have any rules.
a combination is a way of selecting items from a collection, such that (unlike permutation)
the order of selection does not matter. In smaller cases it is possible
to count the number of combinations. For example, given three fruits,
say an apple, an orange and a pear, there are three combinations of two
that can be drawn from this set: an apple and a pear; an apple and an
orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S.
FORMULA FOR COMBINATION
Example 1:
In a lucky draw chits of ten names are out in a box out of which three are to be taken out. Find the number of ways in which those three names can be taken out.
Solution: The possible number of ways for finding three names out of ten from the box is:
C (10, 3) = 10!(3!7!)
=
= 120
So there are 120 different ways of choosing three names out of the ten from the box.
Example 2:
Let us suppose we have 12 adults and 10 kids as an audience of a certain show. Find the number of ways the host can select three persons from the audiences to volunteer. The choice must contain two kids and one adult.
Solution: As order here does not matter so we have:
C (10, 2) * C (12, 1) = [10 * 92
] * [
] = 45 * 12 = 540.
So there are 540 ways in which the host can choose the volunteers containing two kids and an adult.
Example 3:
Let us suppose we have 12 adults and 10 kids as an audience of a certain show. Find the number of ways the host can select three persons from the audiences to volunteer. The choice must contain two kids and one adult.
Solution: As order here does not matter so we have:
C (10, 2) * C (12, 1) = [10 * 92
] * [
] = 45 * 12 = 540.
So there are 540 ways in which the host can choose the volunteers containing two kids and an adult.
Below is the video for combination. 👇
| Exercise | |||||||||||||||||||||||||||
| 1) what is 8! |
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