Permutation and Combination

Permutation and combination are ways to count things, and they help us solve many kinds of problems. A permutation is like arranging your toys on a shelf the order matters. If you put Teddy first, Robot second, and Car third, that’s different from putting Robot first and Teddy second.


A combination is like choosing a group of friends to play with the order doesn’t matter. Whether you say Sam, Mia, Leo, or Leo, Mia, Sam, it’s still the same group of three friends! Let’s explore more about permutations and combinations in a fun and straightforward way below.
 

The methods used to count the number of possible outcomes in a situation are permutations and combinations. Factorials are an important concept used to understand the concepts of permutations and combinations. The product of the first n natural numbers is n! 

Permutations 

Permutations are arrangements of a definite order of a number of events. It is the number of ways a set of objects can be arranged. In permutation, order and the arrangement of data is important. There are two types of permutation:

•  Permutations with repetition: The arrangement of 'r' items from the total of 'n' items, such that each item can be selected more than once, is evaluated by the formula of permutation with repetition.
  
  The formula can be given as : n^k
  
   
•  Permutations without repetition: When the repetition of items are not allowed, then the total number of arrangement is given by permutation without repetition.
  
  The formula for such arrangement is given by: \(P(n,r) = {n! \over (n-r)! \space \times \space r!}\)

Combinations

In combination the number of ways an event can be arranged and here the order doesn't matter. Here, the types of the same kind of things are sorted. In combination, the order and the arrangement of the data do not matter.

Permutation and combination formulas show us how many ways we can arrange items or make groups from a set of things.
 

Use a permutation when the order matters, like deciding who stands first, second, or third in a line.

Use a combination when the order doesn’t matter, like picking three friends to play a game together.

With the same number of items, there are always more permutations than combinations, because changing the order creates extra possibilities.

\(n! = n \times (n - 1) \times \cdots \times 3 \times 2 \times 1 \)
 

\(nCr = \frac{n!}{(n-r)! \, r!} \)
 

\( nPr = \frac{n!}{(n-r)!} \)

The counting problem is examined to determine whether to use permutations or combinations, and the corresponding formula is then applied accordingly.
 

Formula 1: Factorial of a number (n!)

A factorial is the product of all the whole numbers from 1 to n.
It’s like lining up numbers in a row and letting them all join hands!

\(n! = 1 \times 2 \times 3 \times 4 \times \cdots \times n \)
 

Formula 2: Permutations When Order Matters

This formula helps us find how many different arrangements we can make when we pick r things from n things, and the order matters.
Think of it like arranging friends in a line who stands first, second, or third makes a big difference!

\({}^nP_r = \frac{n!}{(n-r)!}, \quad \text{where } 0 \le r \le n \)


Formula 3: Permutations When Repetition Is Allowed

This formula is used when you can choose the same item multiple times. It’s like picking numbers for a secret code you can use the same number again and again!
  

Formula 4: Permutations When Some Items Repeat

This formula is used when you are arranging things, but some of them are the same.
It’s like arranging the letters in the word MOM the two M’s look the same, so swapping them doesn’t make a new arrangement.

\(\frac{n!}{p_1! \, p_2! \, \cdots \, p_k!} \)

 

Formula 5: Combinations When Order Doesn’t Matter

This formula helps us find how many different groups we can make when we pick r items from n items, and the order doesn’t matter.

\({}^nC_r = \frac{n!}{r!(n-r)!}, \quad \text{where } 0 \le r \le n \)

Permutations and combinations are used for counting arrangements and groups. Permutations are used when the order of items matters, as when we line up our toys or arrange participants in a race. Combinations are used when order does not matter, guiding us in choosing friends for a team or picking fruits for a delightful snack.


Permutation (Order Matters): Use permutation when the order of things is important. It’s like making a special line or sequence. For example:

• Lining up your toys from tallest to shortest.
   
• Deciding who stands first, second, and third in a race.
   
• Choosing a password or secret code  the order of numbers or letters matters!
  
   

Combination (Order Doesn’t Matter): Use combination when the order doesn’t matter, and you just want to make a group. For example:

• Picking 3 friends to play a game, it doesn’t matter who you pick first.
   
• Choosing fruits for a snack the order you pick them doesn’t change the group.
   
• Selecting prizes from a treasure box what matters is which prizes you get, not the order.
   

Now let’s learn a few differences between permutations and combinations

The permutation formula is \(P(n,r) = {n! \over (n-r)!}\)

The combination formula is \(C(n,r) = {n! \over (n-r)! \space \times \space r!}\)

 Let us now learn, how these formulas for permutations and combinations are obtained. 

Derivation of Permutation Formula

Permutation is the arrangement of objects or events in an ordered manner. To obtain the formula for permutation, we will consider the given scenario.

When selecting r objects from n objects, the first object can be chosen in n ways. The second object can be chosen from the remaining (n -1) objects, for the third object we are left with (n - 2) objects.

So, \(P(n, r) = n × (n -1) × (n - 2) × ….. × (n - r + 1)\), let’s consider this as 1.
 

Multiplying and dividing 1 with (n - r)!

⇒ \(P(n,r) = {{[n \times (n-1) \times ....... \times (n-r+1)] \times(n-r)!}\over (n-r)! }\)  

Since, the factorial of (n - r) is:

\((n - r)! = (n - r) × (n - r - 1) ×  (n - r - 2) × ….. 3 × 2 ×  1\)

⇒ \(P(n,r) = {{[n \times (n-1) \times ....... \times (n-r+1)] \times [(n-r) \times (n-r-1)....\times 2 \times1]}\over (n-r) \times (n-r-1)....\times 2 \times1 }\)
 

So, \(P(n,r) = {n! \over (n-r)!}\)

Derivation of Combination Formula 

Combination is the way of selecting r objects without considering order. Now let's see what and how the formula of combination derived 

As we know, the formula for permutation is \(P(n, r) = n! / (n - r)!\), it is for r objects out of n objects with order consideration.

As ordering is not considered in combination, let's remove the ordering. Then the number of ways to arrange the selected r object among themselves is r!, 

Since, C(n, r) is the ratio of the total number of permutations to the number of ways to arrange r different objects.

So, \(C(n,r) = {P(n,r) \over r!}\)

That is, \(C(n,r) = {n! \over (n-r)! \space \times \space r!}\)

Permutation and Combination Worksheet
Permutations and combinations are used to find the possible arrangements and selections of objects. Learning them improves analytical skills and helps solve counting problems easily.


Parents and teachers can help by:

• Explaining the difference between permutation and combination in simple words.
   
• Helping kids memorize key formulas.
   
• Guiding them to identify whether a problem is about arrangement or selection.
   
• Breaking complex problems into smaller, easy-to-solve parts.
   
• Practicing with real-life examples, such as arranging toys or picking snacks, can strengthen understanding and problem-solving skills.
   
• Memorize key formulas like \(nPr= n! / (n−r)! ​ \) and \(nCr= n! / r!(n−r)! ​ .\)
   
• Encouraging kids to play fun games or puzzles that involve arranging and selecting items. 
   
• Reinforcing learning with worksheets and small challenges helps students practice consistently.

Permutations and combinations are the techniques we use in math to calculate the order of events, and it is used in different fields. Let’s learn a few real-life applications of permutations and combinations. 
 

• Scheduling and planning events: To plan events in order, we use permutation that can be performed or to schedule meetings and shifts. 
   
• Permutations are used in cryptography and security to estimate the possible keys and to create a secure code.
   
• In lottery and gaming, we use both permutations and combinations. For picking the set of winning numbers in the lottery, we use combinations.
   
• In cooking, both permutations and combinations are used for mixing the ingredients.
   
• Permutations and combinations are used to determine the number of possible ways people can be seated in meetings, classrooms, or theaters based on specific conditions.