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What are Permutations and Combinations?

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Groupings  
 
Say there are four fruits, mango, apple, orange and banana.

Chubby is asked to take (eat) any two of the fruits.

How many choices does Chubby have in selecting the fruits she wants to eat?

Chubby may take what she wants in any one of the following ways,

Mango, Apple Mango, Orange Mango, Banana Apple, Orange Apple, Banana Orange, Banana

• No Preference to the Order in which the objects are chosen

In thinking of the number of choices available to Chubby, we do not pay any attention to the order in which the fruits are being chosen. We only think of what the fruits chosen are?

We say Chubby can choose a Mango and an Orange.

This happens both when Chubby chooses

  • a Mango first and an Orange next; » (Mango, Orange)
    (Or)
  • an Orange first and a Mango next; » (Orange, Mango)

Since we do not give preference to the order in which the fruits are chosen, we consider both of those to represent the same result (grouping).

Thus there are six possible choices for grouping the 4 fruits taking 2 at a time.

Arrangements  
 
Say there are four letters A, B, C, D.

Chikky is asked to frame all possible words using two letters at a time.

How many words can Chikky form.

The Word Formed by Chikky may be

ABACAD BCBDCD
BACADA CBDBDC

• Preference to the Order in which the objects are chosen

permutations,combinations,quantitative,techniques,methods,operations,research,linear,circular
Since what we are forming are words, AB and BA are to be considered as two different entities, though both of them contain the letters A and B only.

This implies that we are giving preference to the order in which we are choosing the letters.

We say Chikky framed the word

  • AB when she chooses A first and B next; » (AB)
    (and)
  • BA when she chooses B first and A next; » (BA)

Since we give preference to the order in which the letters are chosen, we consider both of those to represent different results.

Thus there are 12 possible words (arrangements) that can be formed using the 4 letters taking 2 at a time.

• Groupings in this case

If we do not give preference to the order in which the letters are chosen, then the words AB and BA would represent the same possibility. Thus we would be having only 6 possibilities.

ABACAD BCBDCD
BACADA CBDBDC

There are 6 possible groups (groupings) that can be formed using the 4 letters taking 2 at a time.

Permutations ≡ Arrangements  
 
"Permutations" is a term used to indicate arrangements. Permutations represents

All the possible arrangements of "n" different objects/things taking "r" at a time.

In thinking of permutations we give preference to the order in which the objects involved are chosen.

The number of permutations implies the number of arrangements. Each of the arrangements is called a permutation.

Illustration

There are four games Tennis, Cricket, Football, Basketball.

A student can choose to play any two games, the first game to be played in the first period and the second to be played in the second period.

The choices available to a student would be

Tennis, CricketTennis, FootballTennis, BasketballCricket, FootballCricket, BasketballFootball, Basketball
Cricket, TennisFootball, TennisBasketball, TennisFootball, CricketBasketball, CricketBasketball, Football

The number of choices available to a student would be equal to 12 which is given by
        "The number of arrangements/permutations of 4 (n) things taking 2 (r) at a time".

Combinations ≡ Groupings/Selections  
 
"Combinations" is a term used to indicate groupings/selections. Combinations represents

All the possible groupings/selections of "n" different objects/things taking "r" at a time.

In thinking of combinations we do not give preference to the order in which the objects involved are chosen.

The number of combinations implies the number of groupings/selections. Each of these selections/groupings is called a combination.

Illustration

There are four games Tennis, Cricket, Football, Basketball.

A student can choose to play any two games.

The number of choices available to a student would be

Tennis, Cricket Tennis, Football Tennis, Basketball Cricket, Football Cricket, Basketball Football, Basketball

The number of choices available to a student would be equal to 6 which is given by
      "The number of groupings/combinations of 4 (n) things taking 2 (r) at a time".

Where does the confusion lie?  
 
In problem solving, one major judgment to be made is whether to apply the concept of permutations or combinations to a certain case.

The rule/principle that should guide your decision making is simple.

Wherever the order in which the objects or things are considered is important, use permutations and where the order need not be considered use combinations.
permutations,combinations,quantitative,techniques,methods,operations,research,linear,circular

  1. Seating people in a row

    We have to find — PERMUTATIONS

    In finding the number of ways in which you can seat say 3 (X Y Z) people in a row we consider XYZ and YZX as different seating arrangements. Isn't it? Though in both the cases the people in consideration are X, Y and Z only.

  2. Choosing a team from a set of players

    We have to find — COMBINATIONS

    In forming the team we would be concerned only about who the members of the team are and not about the order in which we include them in the team. Say M, N, P, Q are members of the team whether we express them as M, N, P, Q or M, P, Q, N ... The order has no importance.

Please don't try to remember just by examples. Use examples as a tool to enable your understanding only.

It would not be possible to visualise all possible examples and list out cases where permutations are to be applied and cases where combinations are to be applied. Working out a number of examples would enable you to clarify the difference between permutations and combinations.

Understanding the difference makes your problem solving task simple. True, it is not as easy as is said.

Author Credit : The Edifier ... Continued Page 2

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