If 2^2 ¹⁵C₂ + 4^2 ¹⁵C₄ + 6^2 ¹⁵C₆ + + 14^2 ¹⁵C₁₄ = 2^a 3^b 5^c , where a, b, c N , then the value of a+b+c is :

Step-by-step solution

Let f(x) = _ r=1 ¹⁵ r^2 ¹⁵C_r x^r . We know that (1+x)¹⁵ = _ r=0 ¹⁵ ¹⁵C_r x^r . Differentiating with respect to x and multiplying by x gives: x 15(1+x)¹⁴ = _ r=1 ¹⁵ r ¹⁵C_r x^r Differentiating again with respect to x and multiplying by x gives: 15x(1+x)¹⁴ + 210x^2(1+x)¹³ = _ r=1 ¹⁵ r^2 ¹⁵C_r x^r Thus, f(x) = 15x(1+x)¹⁴ + 210x^2(1+x)¹³ . The sum of the even terms of the series is given by f(1) + f(-1) 2 . Evaluating f(1) : f(1) = 15(1)(2)¹⁴ + 210(1)(2)¹³ = 30 2¹³ + 210 2¹³ = 240 2¹³ f(1) = 15 16 2¹³ = 15 2¹⁷ Evaluat