Introduction to Permutations | Mathematics (Maths) for JEE Main & Advanced PDF Download

What is Permutations and Combinations?

Imagine you go to an ice cream parlor, and they have five delicious flavors: Vanilla, Chocolate, Strawberry, Mint, and Caramel. Now, you want to order an ice cream cone with three different flavors.

Permutations come into play when the order in which you choose the flavors matters. In other words, if you want to create a unique sequence of flavors on your cone, we're talking about permutations.

Here's the deal: for each of your three scoops, you have five flavors to choose from. As you choose each flavor one after the other, the order in which you pick them creates a unique ice cream. That's where the magic of permutations steps in – the art of arranging items in a specific order. 

Now, What if you didn't care about the order, but simply told the person in the ice cream parlour to make an ice cream for you. As the person adds one flavour after the other in any order, it creates different combinations. 

Ice Creams

So with this understanding let's get ready to turn everyday decisions into thrilling mathematical adventures!

Fundamental Principle of Counting

The fundamental counting principle is a rule used to count the total number of possible outcomes in a situation. 

It states that if there are �n ways of doing something, and �m ways of doing another thing after that, then there are �×�n×m ways to perform both of these actions. In other words, when choosing an option for �n and an option for �m, there are �×�n×m different ways to do both actions. 

It would be easier to explain the same using some examples:

Example 1: Planning a Weekend Menu

Imagine you're planning a delightful weekend menu, and you have 2 choices for appetizers (A1, A2) and 3 choices for main courses (M1, M2, M3). Additionally, there are 2 dessert options (D1, D2). How many unique ways can you plan your weekend menu?

Applying the Fundamental Principle of Counting, the number of ways to choose an appetizer is 2, followed by 3 ways to choose a main course. Finally, for each combination, there are 2 options for dessert. Therefore, the total number of ways is 2 (appetizers) × 3 (main courses) × 2 (desserts) = 12 unique weekend menu plans. Easy right? If it’s still not clear, let’s see another one.

The flowchart below depicts this technique for a better understanding:
Planning a Weekend Menu

Example 2: Building a Custom Computer

Suppose you're assembling a custom computer, and you have 4 choices for processors (P1, P2, P3, P4) and 3 choices for graphics cards (G1, G2, G3). Additionally, you can select from 2 types of RAM (R1, R2). How many different configurations can you create for your computer?

Using the Fundamental Principle of Counting, the number of ways to choose a processor is 4, followed by 3 ways to choose a graphics card, and finally, 2 ways to choose RAM. Therefore, the total number of unique computer configurations is 4 (processors) × 3 (graphics cards) × 2 (RAM) = 24.

In both examples, we applied the Fundamental Principle of Counting to calculate the total number of outcomes by multiplying the possibilities for each event. This principle proves to be a versatile and essential tool for systematically exploring and quantifying various sequential events.

Lets see some trickier examples now:

Example 3: Creating Unique Door Sign Combinations

Suppose you have 5 distinct colors (red, blue, yellow, white and black), and you want to create a color by mixing any 2 colors selecting one at a time, without the repetition of color. In how many different ways can you select the two out of given colors?

Using the Fundamental Principle of Counting, 

First Choice: You have 5 options for the first color.

Second Choice: After choosing the first color, you have 4 options left for the second color. (Since you can't repeat the same color, you have one less option.)

Total Ways to Select Two Colors:
To find the total number of ways to select two colors, multiply the number of options at each stage:
Total Ways=Number of options for the first choice×Number of options for the second choice
Total Ways=5×4
So, there are 5×4=20 different ways you can select two colors out of the given five distinct colors. This applies the fundamental principle of counting by considering the independent choices at each stage and multiplying the possibilities.

Therefore, you can create 20 different door sign combinations by pairing two symbols, one above the other. This illustrates how the Fundamental Principle of Counting efficiently determines the various possibilities when events occur sequentially.

Example 4: Designing Unique Bookmarks with Different Features

Suppose you have 6 distinct features (F1, F2, F3, F4, F5, F6), and you want to design unique bookmarks with varying numbers of features. A bookmark can consist of either 2 features, 3 features, 4 features, 5 features, or 6 features, without any repetition. How many different bookmarks can you create?

To determine the total number of possible bookmarks, we'll count the combinations for each scenario separately and then sum the results.

• For 2-feature bookmarks, there are 6 choices for the first feature and 5 choices for the second feature (as we can't repeat features). By the Multiplication Principle, the number of 2-feature bookmarks is 6 × 5 = 30.
• For 3-feature bookmarks, there are 6 choices for the first feature, 5 choices for the second, and 4 choices for the third. The number of 3-feature bookmarks is 6 × 5 × 4 = 120.
• Continuing in the same manner:
  • The number of 4-feature bookmarks is 6 × 5 × 4 × 3 = 360.
  • The number of 5-feature bookmarks is 6 × 5 × 4 × 3 × 2 = 720.
  • The number of 6-feature bookmarks is 6 × 5 × 4 × 3 × 2 × 1 = 720.

Therefore, the total number of unique bookmarks is the sum of these counts: 30 (2-feature) + 120 (3-feature) + 360 (4-feature) + 720 (5-feature) + 720 (6-feature) = 1950.

The flowchart below depicts this technique for a better understanding:
Unique Bookmarks

Permutations

Permutations involve arranging distinct objects in a specific order. Consider the word "ROSE," where each arrangement, such as ROSE, REOS, and so on, is unique. The order of the letters matters in permutations.
For instance, if we want to determine the number of 3-letter words that can be formed from the letters of the word "NUMBER" without repetition, we are counting the permutations of 6 different letters taken 3 at a time. Using the multiplication principle, the total number of arrangements is 6 × 5 × 4 = 120.
If repetition of letters was allowed, the count would be 6 × 6 × 6 = 216.

Definition: A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

Theorem 1: The number of permutations of n different objects taken r at a time (where 0<r≤n) without repetition is denoted as ⁿP_r and is given by: n×(n−1)×(n−2)×…×(n−r+1).

Proof: There are as many permutations as ways of filling r vacant places with n objects. The first place can be filled in n ways, the second in (n−1) ways, and so on. Therefore, the number of permutations is n×(n−1)×(n−2)×…×(n−r+1).

To simplify the notation, we introduce the factorial symbol n! representing the product of the first n natural numbers. For example,
5!=5×4×3×2×1.

Factorial Notation:
n!=1×2×3×…×n. We define
0!=1.
So,

Permutations when all the objects are not distinct 

When dealing with permutations of a set where some objects are not distinct, the counting becomes nuanced. Let's consider an example to illustrate this concept more effectively.

Example: Rearranging the Letters in the word "BOOK"

Suppose we want to find the number of ways to rearrange the letters in the word "BOOK." Here, the letters are not all distinct since there are two 'O's. To tackle this, let's temporarily treat the 'O's as different, labeling them O1 and O2.

The number of permutations of the 4 distinct letters (B, O1, O2, K) taken all at a time is given by 4! (factorial), resulting in 4 × 3 × 2 × 1 = 24 permutations. Consider one of these permutations, say BO1O2K.

However, we have to account for the fact that the 'O's are indistinguishable in the original word. For each arrangement, there are 2! (factorial) ways to rearrange the 'O's amongst themselves without changing the overall permutation. In the example, RO1O2T corresponds to the same arrangement as RO2O1T when the 'O's are not treated as distinct.

To find the actual count of distinct arrangements when the 'O's are considered identical, we divide the total permutations by the number of ways the 'O's can be rearranged among themselves:

Therefore, when arranging the letters in the word "BOOK," considering the 'O's as identical, there are 12 distinct permutations. This approach ensures that we account for the repeated elements properly and avoid overcounting.