COMBINATIONS

 COMBINATIONS 

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter.

Number of combination of N distinct things taken R at a time is given by

r is less than equal to n and n,r are whole numbers.

If n<r than C(n,r)=0

ILLUSTRATIONS 

1)How many diagonals are there in a convex 8 sided polygon ?

Two vertices can be selected from 8 vertices in

C(8,2) = 8!/(8-2)!2!=8!/2!6!=28(Answer)

2) 9 books are arranged in a line on a book shelf in how many ways can we select 3 books such that no consecutive book from the shelf at chosen ?

Let represent the book not selected by 0

So we have .0.0.0.0.0.0. in succession. ( . Represents the spaces where the books can be selected)

The spaces between the zeros can be filled by the books chosen.

So we have 7 spaces to be filled by 3 books.

So we have C(7,3) (Answers)

3)An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is _____  JEE ADVANCE 2020

Use the same idea of Q2 to solve it.

4) A bag contains 2 white balls 3 blue balls and 4 red balls in how many ways 3 balls can be drawn from the bag if at least 1 black ball is to be included in the draw ?

Let W represent white ball

B represent blue ball

R represent red ball

Then we have the following possibilities 

BBB, BBW, BBR , BWW, BRR, BWR

Now we can simply use combination formula 

So we have 

C(3,3) + C(3,2)×C(2,1) + C(3,2)×C(4,1) + C(3,1)×C(2,2) + C(3,1)×C(4,2) + C(3,1)×C(2,1)×C(4,1)

=64

Alternatively this question can be solved with considering only 3 cases -

All blue ball

2 blue ball 1 non blue ball

1 blue ball 2 non blue ball

Many people remember huge number of formula for solving combinatorics problems but using case studies can help us to create our own formula. 

For combinations with repeatation look at our video.