Statistics - Combinations

 
   
     
       

Combinations

       

Understanding Combinations in Mathematics

       

          A combination is a selection of items from a larger set where the order of selection does not matter. It is used to count the number of ways to choose m elements from a total of n distinct elements.        

        Combinations Formula      
     
       
         

Formula for Combinations:

         
           

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

         
         

Where:

         
               
  • \( n \) = total number of items
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  • \( m \) = number of items to choose
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  • \( ! \) = factorial (e.g., \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \))
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Example:

         
           

              \[               C(5, 2) = \frac{5!}{2!(5 - 2)!} = \frac{5!}{2! \cdot 3!}               = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 3 \cdot 2 \cdot 1} = 10               \]            

         
         

There are 10 ways to choose 2 elements from a group of 5 without considering the order.

         

Key Properties of Combinations:

         
               
  • \( C(n, 0) = C(n, n) = 1 \)
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  • \( C(n, m) = C(n, n - m) \) (Symmetry property)
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  • \( C(n, 1) = n \)
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  • Used when order does not matter
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Applications of Combinations:

         
               
  • Calculating lottery probabilities
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  • Group formation in sports, teams, or committees
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  • Data sampling in statistics
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  • Choosing sets of items without replacement in probability
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