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  1. Glossary
  2. /Mathematics
  3. /Permutations and Combinations

What are Permutations and Combinations?

In short

A permutation is an arrangement of objects in which the order matters. A combination is a selection of objects in which the order does not matter.

In simple words

The number of permutations of rrr objects chosen from nnn is nPr=n!(n−r)!^nP_r = \dfrac{n!}{(n-r)!}nPr​=(n−r)!n!​. The number of combinations is nCr=n!r! (n−r)!^nC_r = \dfrac{n!}{r!\,(n-r)!}nCr​=r!(n−r)!n!​. Each combination can be arranged in r!r!r! ways, so nPr=nCr×r!^nP_r = {}^nC_r \times r!nPr​=nCr​×r!. If the question asks for a team, a group or a selection, use combinations. If it asks for a code, a queue or a position, use permutations.

Example

Three medals for 5 runners: 5P3=5×4×3=60^5P_3 = 5 \times 4 \times 3 = 605P3​=5×4×3=60 ways. A team of 3 chosen from the same 5 runners: 5C3=10^5C_3 = 105C3​=10 ways. Also 10×3!=6010 \times 3! = 6010×3!=60.

Also searched as: permutations of distinct objects, 10*9*8.

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Related terms

  • Probability
  • Set

← All glossary terms

Learn it in

  1. The Fundamental Principle of CountingClass 11 to 12
  2. PermutationsClass 11 to 12
  3. CombinationsClass 11 to 12
All lessons on this topic →

Practise

  1. In a round-robin tournament every two teams play each other exactly once, and every…Advanced
  2. A 7 × 7 board has its rows numbered 0 to 6 from the top and its columns numbered 0 to 6…
Advanced
  • A small town has a rectangular grid of streets, shown in the figure…Hard
  • How many 4-letter words, with or without meaning, can be made from the letters of…Medium
    • All problems on this topic →

    Quick questions

    1. Each statement below is about points or whole numbers chosen in any way at all, and…Advanced
    2. The figure shows a regular polygon with 12 vertices, all on one circle. Any three of the…Advanced
    3. Four married couples, eight people in all, sit at a round table with eight equal chairs.…Advanced

    More Mathematics terms

    1. Algebraic Identity
    2. Arithmetic Progression
    3. Complex Number
    4. Derivative
    5. Function
    6. Isosceles Triangle
    All Mathematics terms →

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