Permutation and Combination

Shortcuts - Permutation and Combination

Formation of Numbers with Given Digits:


In these types of questions, it is asked to from numbers with some different digits. These digits can be used with repetetion or without repetetion.

Example: How many 4 digit numbers can be formed with the digits 1, 2, 3, 4 and 5? (Repetetion of digits is not allowed.)

Solution: Required ways = 5 × 4 × 3 × 2 × 1 = 120
Example: How many 4 digit numbers can be formed with the digits 2, 3, 4 and 5? (Repetetion of digits is allowed.)

Solution: Required ways = 4 × 4 × 4 × 4 × 4 = 45
Example: How many numbers between 400 and 1000 can be formed with the digits 2, 3, 4, 5, 6 and 0?

Solution: Here nothing is specified about repetetion of digits, therefore we assume that repetetion of digits is not allowed.
For any number between 400 and 1000 it has to be of 3 digits with hundreds place as 4, 5, 6 (If it starts from 2, 3, 0 then it will not be between 400 and 1000). So, 3 ways for hundreds place.
Tens place can be filled in 5 ways and units place can be filled in 4 ways.
Hence, required ways = 3 × 5 × 4 = 60

Download: Seating Arrangement around Geometrical Figures

Fast Track Formulas/Short Tricks:


Number of circular permutations of n different objects = (n - 1)!

Example: Number of ways in which 5 girls can form a ring?

Solution: Required ways = (5 - 1)! = 4!
In circular permutations, if clockwise and anti-clockwise arrangements are considered to be same, then Number of circular permutations of n objects =

Example: Find the total number of ways, in which 9 beads can form a necklace?

Solution: Required ways =

Let there be 'n' persons in a hall. If every person shakes his hand with every other person only once, then total number of handshakes = C(n,2).

If in place of handshakes each person gives a gift to another person, then the formula is 2 × C(n,2) = n(n - 1)
Number of diagonals in a polygon of 'n' sides = C(n,2) - n

Example: In a party, every person shakes his hand with every other person only once. If total number of handshakes are 210, then find the number of persons?

Solution: Let the number of persons be 'n'. Then C(n,2) = 210.
n = 21.

Download: Permutation and Combination Practice Questions

If there are 'n' non-collinear points in a plane, then
1) Number of straight lines formed = C(n,2)
2) Number of triangles formed = C(n,3)
3) Number of quadrilaterals formed = C(n,4)

Example: In a plane there are 16 non-collinear points. Find the number of straight lines formed?
Solution: Number of straight lines formed = C(16,2) = 120
If there are 'n' points in a plane out of which 'm' are collinear, then
1) Number of straight lines formed = C(n,2) - C(m,2) + 1
2) Number of triangles formed = C(n,3) - C(m,3)

Example: In a plane there are 11 points out of which 5 are collinear. Find the number of triangles formed?
Solution: Number of triangles formed = C(11,3) - C(5,3) = 165 - 10 = 155

Hard Questions:


Question:
In an exam, there are three sections of maximum marks 100. In how many ways can one score 230 marks if in each section, he should get atleast 50 marks?


Required ways = 1926

In case of doubts in any of the questions

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