Addition and Multiplication Principles for JEE

We have natural numbers that are formed for counting from ancient ages, and the two operations were defined on them; addition and multiplication for counting. Here we will discuss the principles of counting addition and multiplication. This is a basic concept of Algebra that further extends to the fundamentals of Permutation. Principles of addition and multiplication are used in counting and that extends to Permutation and then we have the factorial that helps us count objects that are in a particular order.

Addition Principles of Counting

The Addition  Rule asserts that if a task can be completed in two ways but not simultaneously, then the addition of the ways to do the task can be used to complete the assignment. Let, A is an event happening in m ways and B be an event happening in n ways and both the event A and B is independent of each other then A or B any event can happen in m+n ways.

• Let’s give one example for better understanding. Suppose there are 15 NITs and 20 IITs in India and you qualified for the JEE, so there are (15 + 20) = 35 ways you can select your college.

• Another example, consider you go to a restaurant where you find 8 Chicken items and 5 veg items, now you want to order a dish among these. So, you can choose a dish in (8 + 5) = 13 ways. 

Now see in both the examples your choice of one type is independent of the other so, you can just add them together to get total options.

General Addition Principle

So, in general, if you have n disjoint events, i.e. A₁, A₂,..... A_n with outcomes n₁, n₂,.....n_n ways respectively, then any of the events can occur with n₁ + n₂ +..... + n_n ways. 

Multiplication Principle of Counting

The multiplication rule asserts that if a task can be finished by the multiplication of the way the work is completed, then the task may be completed in a sequence of activities one after the other. Suppose A and B are events with n₁ & n₂ possible outcomes, respectively. Then the total number of outcomes for the sequence of the two events is $n_1\times n_2$.

• Let us explain the principle of multiplication with an example: Suppose on Teachers’ day you want to visit your kindergarten, High school, and your college. And there are three ways to connect your kindergarten and High school and five ways to connect your high school and college. Then you can go from your kindergarten to your college in $3\times 5=15$ ways because you can choose any of the three routes for your high school and then again you can choose any of the five routes to your college. Selecting the entire route is dependent on both the routes you choose, hence multiplication.

• Another example is, Suppose there are 20 IITs, each with 7 branches. Then the topper can select his branch in 

$20\times 7=140$ ways.

What Does Factorial Notation Mean?

Suppose we need to organize 2 objects in 2 places. Now for doing this, just like the multiplication example given above 1st we set an object at a place and then set another one at another place. So, A object is set to place “a” and B object is set to place “b” but it can happen if we set A at “b” and B at “a”. Now where the 2nd object is set is solely dependent on where the 1st one is. Now there are 2 ways the 1st object can be placed and that leaves one way for the 2nd object to be set. So, a total number of ways are $2\times 1=2=2!$. So, 2! is basically: in how many ways 2 objects can be placed in two places.

We can generalize this for any $n\in \mathbb{N}$,

Suppose there are “n” cups and “n” balls are to be placed in those cups such that each cup has one ball.

So, for the 1st ball, it has “n” many options to be set and we choose one and set it there. For the next ball, there are “n-1” options to be set and again we choose a place and set it there. We can see for each ball going onwards like this one option decreases and for the nth ball there is only one option. So, the total number of ways is, $n \times(n-1) \times \ldots \times 3 \times 2 \times 1=n !$

So, n! is basically: in how many ways “n” objects can be placed in “n” places.

Factorial Notation

Factorial or “!” is an operation performed on natural numbers and n! or factorial of n means the product of all the natural numbers less than or equal to n.

So, $n !=n \times(n-1) \times(n-2) \times \ldots \times 3 \times 2 \times 1=\prod_{i=1}^{n} i$

1. $1 !=1 \times 1=1$

2. $2 !=1 \times 2=2$

3. $3 !=1 \times 2 \times 3=6$

4. $4 !=1 \times 2 \times 3 \times 4=24$

5. $5 !=1 \times 2 \times 3 \times 4 \times 5=120$

Factorial Notation Properties

Here the properties related to Factorial Notation have been told, which are described below through some points; 

• It is defined as $1!=1$.

• Factorials of negative numbers are not possible because there is no lower bound of numbers. Consider $(-1)!$, there are infinitely many integers less than -1, so the product is not defined.

• We have,

$\begin{align} &n !=n \times(n-1) \times \ldots \times 3 \times 2 \times 1 \\ &\Rightarrow n !=n \times(n-1) ! \end{align}$

• For a $n\in \mathbb{N}$,

$\begin{align} &(2 n) !=1 \times 2 \times 3 \times 4 \times \ldots \ldots \times(2 n-1) \times(2 n) \\ &\Rightarrow(2 n) !=(1 \times 3 \times 5 \times \ldots . . \times(2 n-1)) \times(2 \times 4 \times 6 \times \ldots \times(2 n)) \\ &\Rightarrow(2 n) !=(1 \times 3 \times 5 \times \ldots . . \times(2 n-1)) \times 2^{n} \times(1 \times 2 \times 3 \times \ldots \times n) \\ &\Rightarrow(2 n) !=(1 \times 3 \times 5 \times \ldots . . \times(2 n-1)) \times 2^{n} \times n ! \end{align}$

We can use this formula directly.

Conclusion

This is the main key concept of understanding Combinatorics. If two events occur we 1st see the dependency of events, if they are independent we strictly follow the principle of addition and if they are just dependent then we try the principle of multiplication. As you might have seen, factorial notation is basically a result of multiplication principles, but it is very useful to remember the above-mentioned properties of factorial notation as they come in handy while solving questions for JEE.