Principle Of Mathematical Induction

=

Now,
K = (1 + 2)¹⁰ – 2¹⁰
K = ¹⁰C₀ + ¹⁰C₁ 2 + ¹⁰C₂ 2³ + … + ¹⁰C₁₀ 2¹⁰ – 2¹⁰
K = ¹⁰C₀ + ¹⁰C₁ 2 + 6λ + ¹⁰C₉⋅ 2⁹
K = 1 + 20 + 5120 + 6λ
K = 5136 + 6λ + 5
K = 6μ + 5
Where λ, μ ∈ N
∴ remainder = 5

So, the correct option is (D): 5

So, the answer is .