Question

Evaluate the following using binomial theorem: 

(i) (101)⁴ 

(ii) (999)⁵

Answer 1

(i) (x + a)ⁿ = nC₀ xⁿa⁰ + nC₁ x^n-1 a¹ + nC₂ x^n-2 a² + … + nC_rx^n-r a^r + … + nC_n aⁿ 

(101)⁴ = (100 + 1)⁴ = 4C₀ (100)⁴ + 4C₁ (100)³ (1)¹ + 4C₂ (100)² (1)² + 4C₃ (100)¹ (1)³ + 4C₄ (1)⁴ 

= 1 x (100000000) + 4 x (1000000) + 6 x (10000) + 4 x 100 + 1 x 1 

= 100000000 + 4000000 + 60000 + 400 + 1 

= 10,40,60,401

(ii) (x + a)ⁿ = nC₀ xⁿa⁰ + nC₁ x^n-1 a¹ + nC₂ x^n-2 a² + … + nC_rx^n-r a^r + … + nC_n aⁿ  

(999)⁵ = (1000 – 1)⁵ = 5C₀ (1000)⁵ – 5C₁ (1000)⁴ (1)¹ + 5C₂ (1000)³ (1)² – 5C₃ (1000)² (1)³ + 5C₄ (1000) (1)⁴ – 5C₅ (1)⁵ 

= 1(1000)⁵ – 5(1000)⁴ – 10(1000)³ – 10(1000)² + 5(1000) – 1 

= 1000000000000000 – 5000000000000 + 10000000000 – 10000000 + 5000 – 1 

= 995009990004999