Let S = _ r=1 ¹⁰ r^2 ¹⁰C_r 2^r . If S = 3^ , where and are positive integers and is not divisible by 3 , then the value of + is :

Step-by-step solution

The given series is S = _ r=1 ¹⁰ r^2 ¹⁰C_r 2^r . We can rewrite r^2 as r(r-1) + r . S = _ r=1 ¹⁰ (r(r-1) + r) ¹⁰C_r 2^r S = _ r=2 ¹⁰ r(r-1) ¹⁰C_r 2^r + _ r=1 ¹⁰ r ¹⁰C_r 2^r Using the binomial identities r(r-1) ^ n C_r = n(n-1) ^ n-2 C_ r-2 and r ^ n C_r = n ^ n-1 C_ r-1 : S = _ r=2 ¹⁰ 10 9 ⁸C_ r-2 2^r + _ r=1 ¹⁰ 10 ⁹C_ r-1 2^r Adjusting the powers of 2 to match the lower indices of the combinations: S = 90 2^2 _ r=2 ¹⁰ ⁸C_ r-2 2^ r-2 + 10 2 _ r=1 ¹⁰ ⁹C_ r-1 2^ r-1 Using the expansion of (1+x)^n at x=2 : S = 360(1+2