Combinations and permutations questions are important topic in the competitive exams. We have included some key concepts of the permutations and combinations over here to practice the concepts. Some important rules and conditions are also included to make it easier for you to learn and solve the permutation and combination type questions.
Permutations: Arrangement of finite numbers of article, they can be taken all at time or some at a time
Think Permutation = Position
Combinations: Group of finite number of articles, they can be taken all or some at a time
Difference between the Permutation and Combination –
Permutation = Arrangement ( There is a particular way for arrangement)
Combination = Groups ( they can arranged in any way)
The concepts of counting numbers
Remember – Tip 1
If one can perform a task in ‘m’ different ways and another task can be done in ‘n’ different ways; then numbers of ways the two task can be performed one after another is ‘m x n’ way. Please note it is assumed the performing 1st task in m ways will not have any bearing on performing task 2.
Remember – Tip 2
If one can perform a task in ‘m’ different ways and another task can be done in ‘n’ different ways; then if any task is done at a time the total number of ways of doing this task is m + n
This takes us to concept of factorial n denoted as n! = 1 x 2 x 3 x 4 …. X(n-1)X^n where n is natural number.
Types of Permutation- Since articles can be arranged either in Linear Fashion or Circular Fashion, there are 2 types of permutation
1. Linear Permutation: When articles are arranged in a row it is called linear permutation and is denoted by
In the above formula ‘n’ is total number of articles and r is selected ones chosen to be arranged, thus n>r or n = r, using the above principle of counting it can be denoted.
2. Circular Permutation: When the articles are arranged in circular fashion.
we will discuss this latter on.
- Circular arrangements of n different things with r articles taken at same time =
- Circular arrangements of n different things with all articles taken at same time =
- Circular arrangements of n different things taken r at a time in one direction only =
- Circular arrangements of n different things taken all at a time in one direction only- Clock wise or anticlockwise =
Linear permutations under certain conditions
In all the condition we have considered ‘n’ number of total artciles or objects and ‘r’ artciles or objects taken at a time
- When a particular article is ALWAYS included in arrangement = , if you will note from the formula it is quite simple , assume when a particular article is always included as out of the group that needs to be arranged, hence you will get n – 1 and r – 1
- When a particular article is NEVER included in arrangement= . Similar as above, but since it doesnot come to arrangement, so it part of n things but not part of r things
- When ‘m’ things ALWAYS come together in arrangement = . The trick to remember this formula is consider m things as a single article, so you are left with n- m artilce, now m articles can be arranged them selves as m!
- When ‘m’ things NEVER come together in arrangement = similar as above difference it is reverse of above, also look into 2nd condition
- When ‘r’ things can be repeated the number of arrangement = , it is quite easy to remember, n x n x n …. (r times)
- When at most r can be selected with repetition allowed then number of arrangement can be as follows remember the difference between this and earlier condition is here max no. is r there it is only r
- The no. of arrangement when AT MOST r things can be selected but can’t be repeated
Arrangement when things are similar (Linear)
Since we are in discussion about positions and arrangement (Permutations) if the objects are similar the number of arrangements will get reduced, like say there are 3 blue balls in assortment of 10 balls. Then number of arrangements will get reduced
- number of arrangement for n dissimilar things taken of which r things are similar =
- When there p similar things, q similar things and r similar things in a group of p+q+r things then they can be arranged in
Combinations= Group(s) of the articles taken one, some or all together
Combination = Selection we are not bothered by arrangements or positions of the articles, it is just selections
The combinations are denoted by
Some important rules to remember
Combinations with conditions
If there are n different items and r items are taken at a time then no. of combinations when
- ‘t’ particular things will always included =
- ‘s’ particular things will always included =
- t particular items are always selected and s item are never selected =