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Published on October 16th, 2012 In category Education | Maths

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Combinations and Permutations

Permutation and combinationCombinations 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 Linear permutation

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.

Equation for permutations

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 permutations
  • Circular arrangements of n different things with all articles taken at same time = Circular permutation with all things at a time
  • Circular arrangements of n different things taken r at a time in one direction only = Circular permutation-One direction only
  • Circular arrangements of n different things taken all at a time in one direction only- Clock wise or anticlockwise = Circular permutations- clockwise or anti-clockwise

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 = Permutation when an article is always included ,  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= Permutation when an article is NEVER Included . 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 = Permutation when m things always come together . 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 = Permutation when m things never come together  similar as above difference it is reverse of above, also look into 2nd condition
  • When ‘r’ things can be repeated the number of arrangement = Permutation formula with repetition , 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 Permutation with repetition for max r items 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 Permuatition

 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 = Permuataions for similar things
  • 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 Permutation of group of similar things

 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

Combination formula

Some important rules to remember

  1. Rule for combinations
  2. Combinations Rule 2
  3. Combination Rule 3

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 = Combination rule for some items to be always included
  • ‘s’ particular things will always included = Combination when an item is never selected
  • t particular items are always selected and s item are never selected = Combination formula when t items are selected and s not selected
 There is more to combinations, the items can be divided into groups that can be equal or unequal, the combinations formulae play a great role in making it simpler.
 
Furthermore we will also cover how one can use the formula of combinations to identify number of line, no. of polygon formations, no. of triangle formations etc