If _ j=0 ¹⁰ _ i=0 ^ j ^ 20+i C_ i = m n ³⁰C₁₀ , where m and n are co-prime positive integers, then the value of m + n is

Step-by-step solution

The inner sum is _ i=0 ^ j ^ 20+i C_ i = ²⁰C₀ + ²¹C₁ + ²²C₂ + + ^ 20+j C_ j We know that ^ n C_ r = ^ n C_ n-r . Thus, the sum can be written as: ²⁰C₂₀ + ²¹C₂₀ + ²²C₂₀ + + ^ 20+j C₂₀ Using the hockey-stick identity ^ n C_ r + ^ n C_ r-1 = ^ n+1 C_ r , this sum simplifies to ^ 21+j C₂₁ , which is equal to ^ 21+j C_ j . Now, the outer sum becomes: _ j=0 ¹⁰ ^ 21+j C_ j = ²¹C₀ + ²²C₁ + ²³C₂ + + ³¹C₁₀ Again, rewriting using ^ n C_ r = ^ n C_ n-r : ²¹C₂₁ + ²²C₂₁ + ²³C₂₁ + + ³¹C₂₁ Applying the hockey-stick identity once m