Home / Aptitude / Permutation / Application of Fundamental Principle of Counting
FUNDAMENTAL PRINCIPLE OF ADDITION
If one job can be completed in x ways and another job can be completed in y independent ways then either of the jobs can be done in ‘x+y’ ways
FUNDAMENTAL PRINCIPLE OF MULTIPLICATION
If one job can be completed in x ways and another job can be completed in y independent ways then both the jobs can be done in ‘x × y’ ways.
Ex1. There are 3 cities A, B, and C. City A and B are connected by 2 roads and city B and C connected by 3 independent roads. how many ways can a person go from A to C?
Sol :To go from A to C, the person has to complete both the jobs i.e. going from A to B and then B to C.
So he can go from A to C in 2 × 3 = 6 ways (Principle of Multiplication)
Ex2. A teacher has 3 different pens and 2 different pencils. In how many ways can he give either a pen or a pencil to one of his students.
Sol: Either a pen or a pencil can be given in 3 + 2 = 5 ways (Principle of addition)
Ex3. A teacher has 3 different pens and 2 different pencils. In how many ways can he give one pen and one pencil to one of his students.
Sol: One pen and one pencil can be given in 3 × 2 = 6 ways (Principle of Multiplication)
Q1. 
Mr Ramesh has 4 pairs of trousers and 6 shirts. how many different outfits does he have? 
A. 10 

B. 12 

C. 24 

D. 20 
4 * 6 = 24 (Principle of multipication)
Q2. 
There are 6 gates to enter into the IG park. In how many ways can a child enter into the park and come out with a different gate? 
A. 36 

B. 216 

C. 24 

D. 30 
The child has 6 options to enter and 5 options to come out.
Hence, 6 * 5 = 30 ways (Principle of Multiplication)
Q3. 
There are 4 entrances and 3 exits in a cinema hall. In how many ways can a service boy enter into the hall and come out? 
A. 12 

B. 64 

C. 11 

D. 24 
4 * 3 = 12 (Principle of Multiplication)
Q1. 
In how many ways can 5 letters be inserted into 2 letter boxes? 
A. 10 

B. 25 

C. 32 

D. 5! * 2! 
The first letter can be inserted into any of the 2 boxes = 2 ways
The second letter can be inserted into any of the 2 boxes = 2 ways
All letters can be inserted in 2 × 2 × 2 × 2 × 2 = 32 ways
(Principle of Multiplication)
Q1. 
There are seven tasks (1, 2, 3, 4, 5, 6, 7) which has to be done by 7 people (A, B, C, D, E, F, and G). Each person can do only 1 task. Task 1 can be done by A or B or C. Tasks 4 and 5 can not be done by either F or G. In how many can the tasks be accomplished? 
A. 1728 

B. 864 

C. 5040 

D. 216 
Task 1 can be done in 3 ways (Either A or B or C)
Task 4 can be done in 4 ways (One of A/B/C is out and F& G out)
Task 5 can be done in 3 ways (One of A/B/C is out. F & G out and one of the rest is also out for task 4)
Other tasks can be done in 4 × 3 × 2 × 1 ways
So total number of ways = 3 × 4 × 3 × 4 × 3 × 2 ×1 =864 ways
Q2. 
6 friends are writing Common Admission Tests for admission into 9 IIMs. In how many ways can their result be achieved if each student can get into atmost 1 IIM? 
A. 6 ^{9} 

B. 9 ^{6} 

C. 54 

D. 1 million 
Each student can achieve 10 results (any 9 IIMs or no IIM)
6 students = 10 ×10 ×10 ×10 × 10 × 10 = 1000000