Statistics - Permutations

 
   
     
       

Permutations

       

Understanding Permutations in Combinatorics

       

          A permutation refers to the number of ways to arrange a subset of items from a larger set where the order matters. It is used when you're forming sequences, rankings, or positions from a group of elements.        

       
          Permutations Formula        
     
     
       
         

Formula:

         
           

              \[               P(n, m) = \frac{n!}{(n - m)!}, \quad (n \geq m)               \]            

         
         

Where:

         
               
  • \(n\): Total number of distinct items
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  • \(m\): Number of items selected (ordered subset)
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  • \(!\): Factorial function (e.g., \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\))
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Example:

         
           

              \[               P(5, 2) = \frac{5!}{(5 - 2)!} = \frac{5!}{3!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1} = 20               \]            

         
         

There are 20 ways to arrange 2 objects from a group of 5 in a specific order.

         

Key Properties of Permutations:

         
               
  • Order of selection is important
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  • Used when no repetition of elements is allowed
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  • \(P(n, n) = n!\)
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  • \(P(n, 0) = 1\): There’s one way to select nothing
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Applications of Permutations:

         
               
  • Arranging people in a line or order of finish in a race
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  • Generating passwords or unique access codes
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  • Ranking candidates, players, or participants
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  • Solving problems involving scheduling and sequencing
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