Solving Problems with Combinations Formula

Fundamental Principle of Counting:

According to the fundamental principle of counting, if you have to determine the number of arrangements of two events simultaneously, it is the product of the number of individual events. 

Suppose, if an event happens x number of times and another event occurs y number of times, then the total possible ways of arrangement can be found by x × y.

Multiplication Principle:

The multiplication principle extends the fundamental principle of counting. When several events co-occur, the total number of possible arrangements is the product of the number of each event.

For example, if an event occurs m1 number of times following which a second event occurs m2 times and a third event m3 times and so on till an event occurs mn number of times, then the total number of arrangements is m1× m2×…×mn.

Permutations:

This is one of the counting techniques that help determine the number of different ways of arranging a particular group. 

Permutations are an arrangement of a group of numbers in a fixed order. You can take some or all numbers at a time. 

If repetition is not permitted, the formula for permutations stands as

nPr = n! / (n – r)!

Where,

n = total number of different objects

r = chosen object

n! = n x (n-1) x (n-2) x (n-3) x …x 1

Permutation, when repetition is allowed, is given by nr.

So, before we begin combinations, it is important to know the meaning of factorial notation.

What is the Factorial Notation?

The n! pronounced as n factorial is the product of the first n natural numbers. 

n! = 1 x 2 x 3 x…x (n – 1) x n

Combinations:

If you want to select some items from a group without arranging them, solve such problems with combinations.

The combinations formula determines the total possible ways of selecting all objects or some of them from a group without arranging them. 

If you want to select r number of objects from n given objects, then the combination formula is given by,

nCr = n! / r! (n – r)!

How to Choose Combinations and Permutations?

You can use the table below to determine whether the questions represent problems with combinations or permutations. 

Some Important Combinations Formulas: 

You can use the following formulas to solve problems with combinations.

1. nCr = nCn – r

2. nCr + nCr – 1 = n + 1Cr

3. n × n – 1Cr – 1 = (n- r + 1) × nCr – 1

Example of Combinations Formula:

Here is an example for a better understanding of the concept of combinations. The way of solving a factorial is also shown in the solution.

Suppose a student has to answer five questions out of 7. The student must choose at least 1 question from Section A and B. There are three questions in Section A and four questions in Section B. How many ways of selection are possible?

The student can select in the following ways:

3 and 2 questions from Section A and Section B respectively = 3C3 x 4C2

= [3! / 3! (3-3)!] x [4! / 2! (4-2)!]

= {(1 x 2 x 3)/ [(1 x 2 x 3) x 0!]} x [(1 x 2 x 3 x 4)/ (1 x 2) (1 x 2)]

= 1 x 6 = 6

*0! = 1

Similarly,

2 and 3 questions from Section A and Section B respectively = 3C2 x 4C3 = 3 x 4 = 12

1 and 4 questions from Section A and Section B respectively = 3C1 x 4C4 = 3 x 1 = 4

So, the number of ways to select a question is –

3C3 x 4C2 + 3C2 x 4C3 + 3C1 x 4C4

= 6 + 12 + 4

= 22

Here are some of the commonly asked questions about solving problems with combinations.

Conclusion:

Combinations are one of the useful counting techniques to find the number of possible ways of selecting one or several objects in a group. 

You can solve problems with combinations easily by understanding the differentiating factors between combinations and permutations and applying the proper formula. 

The above topic will not only help aspiring IIT/JEE students in the entrance examinations, but it also holds immense value in real life.