Prove that  

Show that the n^th coefficient in the expansion of (1 – x)^{− n}  is double of (n – 1)^th

Find the greatest term in the expansion of (1 + x)⁻ⁿ when .

What is the greatest term in the expansion of  when the value of x is  .

Find the coefficient of xⁿ in the expansion of  

Prove that  

Prove that

Prove that

Prove that

Find the coefficient of x^r in the expansion of

If n is any positive integer show that the integral part of  is an odd number.

Show that the integral part of  is odd, if n is a positive integer.

Find the coefficient of xⁿ in the expansion of
(1 – 2x + 3x² – 4x³+ .  .  .)^–n

Prove that the expansion of (1 – x³)ⁿ may be put in the form
 

Prove that the coefficient of xⁿ in the expansion of  is 1, 0, −1 according as n is of the form 3m, 3m – 1, 3m + 1

If S_n denotes the sum of first n natural numbers prove that
(1 – x)⁻³ = S₁ + S₂x +S₃x² + .  .  . + S_nx^{n – 1}+ .   .  .

If C₀, C₁, C₂ .  .  . C_n are coefficients in the expansion of (1 + x)ⁿ where n is a positive integer, show that
 

If  where n and p are positive integers and β is a proper fraction, show that (1 – β) (p + β) = 1

Find the sum of the products, two at a time, of the coefficients in the expansion of (1 + x)ⁿ, when n is a positive integer

If the expansion  be
+x++ … ++…. +, show that

+++ … = +++ … = +++ … = 3ⁿ⁻¹

If , prove that
+++ … +=