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Mathematics Dictionary
Dr. K. G. Shih
Binomial Theory


  • Q01 | - Deifintion of Binomial Theorem
  • Q02 | - Coefficients of Binomial expansion
  • Q03 | - Coefficients of binomial expansion and Pascal triangle
  • Q04 | - References

    • Q01. Deifintion of Binomial Theorem

    Deifintion of Binomial Theorem
    • (x + y)^n = C(n,0)*x^n + C(n,1)*(x^(n-1))*(y)+ C(n,2)*(x^(n-2))*(y^2) + ....
    • Coefficients of expansion
      • C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/r!.
      • C(n,0) = Coefficient of first term = 1.
      • C(n,n) = Coefficient of last term = 1
      • C(n,1) = Coefficient of 2nd term = n
      • C(n,n-1) = Coefficient of last 2nd term = n
    • It can be expressed as (x + y)^n = Sum[C(n,r)*(x^(n-r))*(y^r)]
    If n is positive integer
    • It has (n + 1) terms
    • Sum of the coeefficients = C(n,0) + C(n,1) + .... = 2^n.
      • Let x = y = 1, we have
      • C(n,0) + C(n,1) + .... + C(n,n) = 2^n.
    • Sum of the coefficients of odd terms = sum of coeff of even terms.
      • Let x = 1 and y = -1, we have
      • C(n,0) + C(n,2) + ... = C(n,1) + C(n,3) + ...
    The power of n
    • It can be positive integer
      • (x + y)^10 = x^10 + 10*(x^9)*y + 10*9*(x^8)*(y^2)/(2!) + .... + y^10
    • It can be negatiive integer
      • (x + 1)^(-1) = x^(-1) + (-1)*(-2)*(x^(-2))/(2!) + ....
      • = 1/x - 1/(x^2) + 1/(x^3) - 1/(x^4) + ....
    • It can be fraction
      • (x + 1)^(1/2) = x^(1/2) + (1/2)*(1/2 - 1)*(x^(-1/2))/2! + ....
      • = x^(1/2) - (1*3)*(x^(-1/2))/((2^2)*(2!))
      • + (1*3*5)*(x^(-3/2))/((2^3)*(3!)) + ...

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    Q02. Coefficients of Binomial expansion

    Coefficients of Binomial expansion
    • C(n,r) is the coefficient (x^(n-r))*(y^r) in (x + y)^n
    • x power + y power = n.
    Example : Find coefficient of (x^5)*(y*4) in expansion of (x + y)^n.
    • n = 5 + 4 and r = 4.
    • Coeficient of (x^5)*(y^4) = C(9,4)
    • = 9*8*7*(9 - 4 + 1)/4!
    • = (9*8*7*6)/(4*3*2*1)
    • = 126.
    Example : Prove that C(n,r) = C(n,n-r)
    • C(n,r)
      • = n*(n-1)*...*(n-r+1)/r!
      • = n*(n-1)*(n-2)*...*(n-r+1)*(n-r)!/((r!)*(n-r)!)
      • = n!/((r!)*(n-1)!)
    • C(n,n-r)
      • = n*(n-1)*(n-2)*...*(n-(n-r)+1)/(n-1)!
      • = n*(n-1)*(n-2)*....*(r+1)/(n-1)!
      • = n*(n-1)*(n-2)*....*(r+1)*r!/((n-1)!)*r!)
      • = n!/((r!)*(n-1)!)
    • Hence C(n,r) = C(n,n-r).
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    Q03. Coefficients of binomial expansion and Pascal triangle

    Pascal triangle : It is coeff of binomial expansion
    • 1, 2, 1 ............. ceefficients of expansion of (x+y)^2
    • 1, 3, 3, 1 .......... ceefficients of expansion of (x+y)^3
    • 1, 4, 6, 4, 1 ....... ceefficients of expansion of (x+y)^4
    • 1, 5, 10, 10, 5, 1 .. coefficients of expansion of (x+y)^5
    Pscal triangle with r
      Column ...... r0 r1 r2 r3 r4 r5 r6 r7 r8
      ---------------------------------
      n=0 ......... 01
      n=1 ......... 01 01
      n=2 ......... 01 02 01
      n=3 ......... 01 03 03 01
      n=4 ......... 01 04 06 04 01
      n=5 ......... 01 05 10 10 05 01
      n=6 ......... 01 06 15 20 15 06 01
      n=7 ......... 01 07 21 35 35 21 07 01
      n=8 ......... 01 08 28 56 70 56 28 08 01
    [Example] Find coefficient of (x^3)*(y^5) in expansion of (x+y)^n
    • The power n = 3 + 5 = 8
    • For y^5 we know r = 5
    • At row n = 8 and r = 5
    • we have C(n,r)
      • = C(8,4)
      • = 8*(8-1)*(8-2)*(8-3)*(8-4)/(5!)
      • = 8*7*6*5*4/(5*4*3*2*1)
      • = 7*2*4
      • = 56
    • This is the number along column r5 at row n = 8
    • Hence the use of Pascal triangle is easier if n is not to large

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    Q4. References

    References :
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