If 26 ( 2^3 3 12 2 + 2^5 5 12 4 + 2^7 7 12 6 + + 2¹³ 13 12 12 ) = 3¹³ - , then is equal to:

Step-by-step solution

Let S = 2^3 3 ¹²C₂ + 2^5 5 ¹²C₄ + 2^7 7 ¹²C₆ + + 2¹³ 13 ¹²C₁₂ Using the property 1 r+1 ^ n C_ r = 1 n+1 ^ n+1 C_ r+1 , we can rewrite the terms: S = 2^3 13 ¹³C₃ + 2^5 13 ¹³C₅ + 2^7 13 ¹³C₇ + + 2¹³ 13 ¹³C₁₃ 13S = 2^3 ¹³C₃ + 2^5 ¹³C₅ + 2^7 ¹³C₇ + + 2¹³ ¹³C₁₃ We know the binomial expansions: (1+x)¹³ = ¹³C₀ + ¹³C₁x + ¹³C₂x^2 + ¹³C₃x^3 + + ¹³C₁₃x¹³ (1-x)¹³ = ¹³C₀ - ¹³C₁x + ¹³C₂x^2 - ¹³C₃x^3 + - ¹³C₁₃x¹³ Subtracting the second equation from the first gives: (1+x)¹³ - (1-x)¹³ = 2 ( ¹³C₁x + ¹³C₃x^3 + ¹³C₅x^5 + + ¹³C₁₃x¹³ )