Binomial Coefficients

1) Prove:

A) ⁿ⁻¹C₁ + ⁿ⁻¹C₂ + ⁿ⁻¹C₃+...+ ⁿ⁻¹Cₙ₋₁ =2ⁿ⁻¹ - 1.

B) ⁿC₀ + ⁿC₂ + ⁿC₄ + ⁿC₈ +... = 2ⁿ⁻¹.

2) If (1+x)ⁿ = C₀ + C₁ₓ + C₂ₓ² + .... + Cₙₓⁿ Show that,

A) C₁+2C₂+3C₃+....+ ₙCₙ= n.2ⁿ⁻¹.

B) C₀ + 3C₁ + 5C₂/3 + ...+

+(2n+1) Cₙ = (n+1) 2ⁿ 

C) C₁- 2C₂+3C₃+...+(-1) ⁿ⁻¹. ₙCₙ=0

D) C₀+4C₁+ 8C₂+12C₃+...+4ₙCₙ = 1+ ₙ. 2ⁿ⁺¹.

E) C₀+2C₁+2²C₂+...+ 2ⁿ .Cₙ = 3ⁿ

F) (C₀ + C₁+C₂+...+Cₙ)²= ²ⁿC₀+ ²ⁿC₁ + ²ⁿC₂ + ...+ ²ⁿC₂ₙ .

3) If (1+x)ⁿ = C₀ + C₁ₓ + C₂ₓ² + .... + Cₙₓⁿ Show that

A) C₀/1 - C₁/2 + C₂/3 -...+(-1) ⁿ.Cₙ/(n+1) = 1/(n+1)

B) C₁/2 + C₃/4 + C₅ +....= (2ⁿ -1)/(n+1)

4) If (1+x)ⁿ= C₀ + C₁ₓ + C₂ₓ² + .... + Cₙₓⁿ Show that,

A) C₀² + C₁² + C₂² + .... + C²ₙ = (2n!)/(n!)².

B) C₀ C₁ + C₁ C₂ + C₂C₃ + ...+ Cₙ₋₁Cₙ = (2n!)/{(n+1)! (n -1)!}

C) C₀ᵣ + C₁Cᵣ₊₁+ C₂ C ᵣ₊₂+...Cₙ₋ᵣCₙ = (2n!)/{(n+r)! (n -r!}

5) (1+x)ⁿ= C₀ + C₁ₓ + C₂ₓ² + .... + Cₙₓⁿ show that,

A) C₁/C₀ + 2 C₂/C₁ +3C₃/C₂ +..... + nCₙ/Cₙ₋₁ = n(n+1)/2

B) (C₀ + C₁)+ (C₁+ C₂) + (C₂+C₃) ... +(Cₙ₋₁+Cₙ)= (n+1)ⁿ/n! C₁C ₂C₃..Cₙ    

6) (1- x)ⁿ= C₀ - C₁ₓ + C₂ₓ² +......+(-1)ⁿxⁿ, show that

C₁ + 2C₂ + 3C₃ +n. Cₙ = n. 2ⁿ⁻¹

7) If n be an integer greater than 1, then show that,

a - ⁿC₁ (a-1) + ⁿC₂ (a-2) -...

(-1)ⁿ (a-n) = 0

8) If (1+x +x²)ⁿ = a₀ + a₁x +a₂x² + .... a₂ₙ x²ⁿ then prove that 

a₀ + a₂ + a₄ + ... a₂ₙ = (1/2) (3ⁿ +1)