Some Results on Binomial Coefficients and Negative or Fractional Indices - Mathematics (Maths) Class 11 - Commerce

D. Some Results on Binomial Coefficients 

(a) C₀ + C₁ + C₂ + ............+ C_n = 2ⁿ

(b) C₀ + C₂ + C₄ + ............= C₁ + C₃ + C₅ + .......... = 2^{n -1}

Remember : (2n) ! = 2ⁿ. n! [1.3.5........(2n -1)]

Ex.8 If (1 + x)ⁿ = C₀ + C₁x + C₂x² +.................+ C_nxⁿ then show that the sum of the products of the  taken two at a time represents by :  

Sol. 

Since (C₀ + C₁ + C₂ + ........+C_{n – 1} + C_n)² = C₀² + C₁² + C₂² + ...... + C_n-1 ²+ C_n² + .......+

2(C₀C₁ + C₀C₂ + C₀C₃ + ... + C₀C_n + C₁C₂ + C₁C₃ + C₁C_n + C₂C₃ + C₂C₄ + ... + C₂C_n + ...... + C_n–1C_n )

Ex.9  If (1 + x)ⁿ = C₀ + C₁x + C₂x₂ + .........+C_nx_n then prove that 

Sol. 

  (C₀ + C₁)2 + (C₀ + C₂)2 + ....+ (C₀ + C_n)² + (C₁ + C₂)² + (C₁ + C₃)² + ....+ (C₁ + C_n)² + (C₂ + C₃)² + (C₂ + C₄)² + ........+ (C₂ + C_n)² + .........+ (C_n–1 + C_n)²

 E. Binomial theorem for negative or fractional indices

If n ∈ Q, then (1 + x)ⁿ =    provided |x| < 1.

Note : 

(i) When the index n is a positive integer the number of terms in the expansion of (1 + x)ⁿ is finite i.e. (n + 1) & the coefficient of successive terms are : ⁿC₀, ⁿC₁, ⁿC₂, .........., ⁿC_n

(ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 + x)ⁿ is infinite and the symbol ⁿC_r cannot be used to denote the coefficient of the general term.

(iii) Following expansion should be remembered (|x| < 1)

(a) (1 + x)^–1 = 1 – x + x² – x³ + x⁴ – .........∝

(b) (1 – x)^–1 = 1 + x + x² + x³ + x⁴ + .........∝

(c) (1 + x)^–2 = 1 –2x + 3x² – 4x³ +  .........∝

(d) (1 – x)^–2 = 1 + 2x + 3x² + 4x³ + .........∝

(iv) The expansions in ascending powers of x are only valid if x is 'small'. If x is large i.e. |x| > 1 then we may find it convenient to expand in powers of 1/x, which then will be small.