What are Permutations and Combinations?

"Permutations" and "Combinations" are two distinct terms used to indicate arrangements and groupings/selections respectively. To understand the difference between the two, let us consider the following illustrations.

• Groupings

  Say there are four fruits, mango, apple, orange and banana. Chubby is asked to eat any two of the fruits. How many choices does Chubby have in selecting the fruits she wants to eat?. Chubby may,

  Mango, Apple

  Mango, Orange

  Mango, Banana

  Apple, Orange

  Apple, Banana

  Orange, Banana

  No Preference to the Order in which the objects are chosen

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  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 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

  AB	AC	AD	BC	BD	CD
  BA	CA	DA	CB	DB	DC

  Preference to the Order in which the objects are chosen

  Since we are forming words, AB and BA are to be considered as two different words, 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 objects.

  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 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.

  Thus there are 6 possible groups that can be formed using the 4 letters taking 2 at a time.

Permutations and Combinations

• Combinations ≡ Groupings/Selections

  Combinations is a term used to represent all 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. Each of these selections/groupings is called a combination.

  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 "The number of groupings/combinations of 4 things taking 2 at a time" which is 6 here.

• Permutations ≡ Arrangements

  Permutations is a term used to represent 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. Each of these arrangement is called a permutation.

  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 number of choices available to a student would be

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

  The number of choices available to a student would be equal to "The number of arrangements/permutations of 4 things taking 2 at a time" which is 6 here.

Why do we get confused with these?

In problem solving, one major problem many face is with judging whether to apply the concept of permutations or combinations to a certain case. The rule/principle is very simple. Wherever the order in which the objects or things are considered is important, we have to use permutations and where the order is not be considered we use combinations.

Understanding the difference makes your problem solving task simple. Keep working on a number of examples to enable you to clarify the difference between permutations and combinations. It would not be possible to visualise all possible examples.

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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.

 

Author Credit : The Edifier

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