Binomial theorem

<p>In <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a>, the binomial theorem (or binomial expansion) describes the <a href="/facts/Polynomial_expansion/oIe5fBKr">algebraic expansion</a> of <a href="/facts/Exponentiation/pac6QY2g">powers</a> of a <a href="/facts/Binomial_(polynomial)/cfFcsR8h">binomial</a>. According to the theorem, the power ⁠
  
    
      
        
          (
          x
          +
          y
          
            )
            
              n
            
          
        
      
    
    {\displaystyle \textstyle (x+y)^{n}}
  
⁠ expands into a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> with terms of the form ⁠
  
    
      
        
          a
          
            x
            
              k
            
          
          
            y
            
              m
            
          
        
      
    
    {\displaystyle \textstyle ax^{k}y^{m}}
  
⁠, where the exponents ⁠
  
    
      
        k
      
    
    {\displaystyle k}
  
⁠ and ⁠
  
    
      
        m
      
    
    {\displaystyle m}
  
⁠ are <a href="/facts/Nonnegative_integer/u65og8JE">nonnegative integers</a> satisfying ⁠
  
    
      
        k
        +
        m
        =
        n
      
    
    {\displaystyle k+m=n}
  
⁠ and the <a href="/facts/Coefficient/5Hwq2Wuq">coefficient</a> ⁠
  
    
      
        a
      
    
    {\displaystyle a}
  
⁠ of each term is a specific <a href="/facts/Positive_integer/u65og8JE">positive integer</a> depending on ⁠
  
    
      
        n
      
    
    {\displaystyle n}
  
⁠ and ⁠
  
    
      
        k
      
    
    {\displaystyle k}
  
⁠.  For example, for ⁠
  
    
      
        n
        =
        4
      
    
    {\displaystyle n=4}
  
⁠,

(
        x
        +
        y
        
          )
          
            4
          
        
        =
        
          x
          
            4
          
        
        +
        4
        
          x
          
            3
          
        
        y
        +
        6
        
          x
          
            2
          
        
        
          y
          
            2
          
        
        +
        4
        x
        
          y
          
            3
          
        
        +
        
          y
          
            4
          
        
        .
      
    
    {\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}

</p><p>The coefficient ⁠
  
    
      
        a
      
    
    {\displaystyle a}
  
⁠ in each term ⁠
  
    
      
        
          a
          
            x
            
              k
            
          
          
            y
            
              m
            
          
        
      
    
    {\displaystyle \textstyle ax^{k}y^{m}}
  
⁠ is known as the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> ⁠
  
    
      
        
          
            
              
                (
              
              
                n
                k
              
              
                )
              
            
          
        
      
    
    {\displaystyle {\tbinom {n}{k}}}
  
⁠ or ⁠
  
    
      
        
          
            
              
                (
              
              
                n
                m
              
              
                )
              
            
          
        
      
    
    {\displaystyle {\tbinom {n}{m}}}
  
⁠ (the two have the same value). These coefficients for varying ⁠
  
    
      
        n
      
    
    {\displaystyle n}
  
⁠ and ⁠
  
    
      
        k
      
    
    {\displaystyle k}
  
⁠ can be arranged to form <a href="/facts/Pascal%2527s_triangle/tb6BKXbc">Pascal's triangle</a>. These numbers also occur in <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, where ⁠
  
    
      
        
          
            
              
                (
              
              
                n
                k
              
              
                )
              
            
          
        
      
    
    {\displaystyle {\tbinom {n}{k}}}
  
⁠ gives the number of different <a href="/facts/Combinations/nmvrhMm1">combinations</a> (i.e. subsets) of ⁠
  
    
      
        k
      
    
    {\displaystyle k}
  
⁠ <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> that can be chosen from an ⁠
  
    
      
        n
      
    
    {\displaystyle n}
  
⁠-element <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>.  Therefore ⁠
  
    
      
        
          
            
              
                (
              
              
                n
                k
              
              
                )
              
            
          
        
      
    
    {\displaystyle {\tbinom {n}{k}}}
  
⁠ is usually pronounced as "⁠
  
    
      
        n
      
    
    {\displaystyle n}
  
⁠ choose ⁠
  
    
      
        k
      
    
    {\displaystyle k}
  
⁠".
</p>

Binomial theorem open-in-new

Binomial theorem