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Permutations and combinations are two basic ideas in mathematics that help us count and arrange things. They are part of a branch of math called combinatorics, which is all about figuring out how many ways we can choose or arrange items from a group.
A permutation is used when the order of items matters. For example, if you're arranging people in a line or assigning positions like first, second, and third, the order changes the result, so we use permutations.
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A combination is used when the order doesn’t matter. For example, if you're choosing toppings for a pizza or selecting team members, it doesn't matter in which order you pick them, so we use combinations.
These concepts are used in many real-life situations like creating passwords, planning events, choosing teams, and even solving puzzles. They are also important in statistics, probability, and computer science.
Table of Contents:
This article aims to provide a comprehensive understanding of permutations and combinations. It covers their definitions, formulas, differences, uses, and some solved examples. It also includes a Permutation And Combination Worksheet for students to practice and enhance their understanding of these concepts.
In mathematics, a permutation means arranging the items of a group in a specific order. When you change the order of things, you are making a new permutation. For example, if you have three letters A, B, and C, then ABC, BCA, and CAB are all different permutations because the order is different in each case. Permutations are useful when the position or sequence of items matters. This concept is often used in problems involving arranging people, numbers, or objects, and is important in subjects like probability, algebra, and logic.
Learn more about Permutation here.
A combination is a way of choosing items from a group where the order doesn't matter. For example, choosing two fruits from apple, banana, and orange — picking apple and banana is the same as picking banana and apple. In combinations, we are only interested in which items are chosen, not how they are arranged. When we select k items out of n without repeating any, it’s called a simple combination. If items can be repeated, it’s called a combination with repetition or k-selection. Combinations are often used in probability, statistics, and everyday choices.
Get more insights about Combination here.
There are several formulas associated with the concepts of permutation and combination. The two fundamental formulas are:
A permutation involves the selection of 'r' items from a set of 'n' items, where the order of selection matters and replacement is not allowed.
nPr = (n!) / (n-r)!
A combination involves the selection of 'r' items from a set of 'n' items, where the order of selection doesn't matter and replacement is not allowed.
Permutation and Combination are important math concepts that help us count the number of ways things can be arranged or selected.
A permutation is used when the order matters. For example, if you are arranging 3 books on a shelf, placing them in different orders counts as different permutations. The formula to find the number of permutations is:
nPr = n! / (n – r)!
A combination is used when the order doesn’t matter. For example, if you are choosing 2 friends from a group of 5 to go on a trip, the order you choose them doesn’t matter. The formula for combinations is:
nCr = n! / [r! × (n – r)!]
These topics are used in probability, statistics, and everyday decisions, like forming teams, passwords, or schedules. In exams like SSC, Banking, and Railways, questions on this topic test your logical and analytical thinking.
To master this topic, it's important to understand whether order is important in the given problem. Once that’s clear, use the right formula and solve with confidence!.
We can understand how the formulas for permutation and combination are formed by using basic counting methods. Let’s break it down simply.
Permutation means choosing and arranging r items out of n total items, where order matters.
Let’s say you want to pick r items from n, one after another:
So, the total number of ways to arrange r items out of n is:
P(n, r) = n × (n − 1) × (n − 2) × ... up to r terms
Now, to write this in a simpler form, we multiply and divide by (n − r)!:
P(n, r) = n! / (n − r)!
Combination means choosing r items from n items, but order does not matter.
From above, we know:
P(n, r) = n! / (n − r)!
But in combinations, all different orders of the same items are considered the same, so we divide by r! (the number of ways to arrange r items):
C(n, r) = P(n, r) / r! = n! / [(n − r)! × r!]
So, the final formula for combinations is:
C(n, r) = n! / [(n − r)! × r!]
Check out the differences between permutation and combination below.
Permutation |
Combination |
Arranging digits, letters, and colors |
Selecting a menu, clothes, subjects, team |
Choosing a team leader and team members from a group |
Selecting team members from a group |
Picking two favorite colors in order from a color palette |
Picking two colors from a color palette |
Picking first, second, and third place winners |
Picking three winners |
Permutation and Combination are powerful mathematical tools used to count, arrange, or select objects in a variety of ways. These concepts are widely used in daily life, academics, and different fields such as business, technology, and science.
Permutations are used when we want to arrange objects and the order of arrangement matters. This is important in cases such as:
Combinations are used when we want to select items or people, but the order of selection does not matter. Common examples include:
In real-life problems, permutation and combination help us figure out the number of possible choices or outcomes. For example:
These concepts are used in:
In biology and chemistry:
Example 1:
Determine the number of permutations and combinations for n = 10 and r = 3.
Solution:
Given n = 10 and r = 3, we can use the permutation and combination formulas as follows:
Permutation:
ⁿPᵣ = (n!) / (n - r)! = (10!) / (10 - 3)! = 720
Combination:
ⁿCᵣ = n! / [r!(n - r)!] = 10! / [3!(10 - 3)!] = 120
Example 2: If the letters of the word "HELLO" are arranged in all possible ways and listed in dictionary order, what is the 49th word?
Solution:
The word "HELLO" has the letters: H, E, L, L, O
Step 1: Arrange the letters in alphabetical order:
E, H, L, L, O
Now we find out how many words start with each letter (like a dictionary):
Start with E:
Remaining letters: H, L, L, O
Total arrangements = 4! / 2! = 12
Words 1 to 12
Start with H:
Remaining letters: E, L, L, O
Total arrangements = 4! / 2! = 12
Words 13 to 24
Start with L:
Remaining letters: E, H, L, O
Total arrangements = 4! = 24
(Because only one L is left to arrange, no repetition)
Words 25 to 48
Now we have listed 48 words so far.
The 49th word will be the first word starting with O.
Remaining letters: E, H, L, L
Arranged in dictionary order: E < H < L < L
So the first word formed using O + E, H, L, L is:
49th word = OEHLL
OEHLL is the 49th word in dictionary order.
Example 3:
In how many ways can a committee of 5 men and 3 women be selected from 8 men and 10 women?
Solution:
Selecting 5 men out of 8 = 8C5 ways = 56 ways
Selecting 3 women out of 10 = 10C3 ways = 120 ways
Total number of ways = (56 x 120) = 6720 ways. So, the committee can be selected in 6720 ways.
Question 1: How many ways can the letters of the word "MATH" be arranged so that all the vowels come together?
Question 2: How many teams of 4 girls and 3 boys can be formed from a group of 8 girls and 9 boys?
Question 3: How many words can be formed by 3 vowels and 4 consonants taken from 5 vowels and 7 consonants?
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