If a r is the coefficient of x 10 - r in the Binomial expansion of 1 + x 10 , then ∑ r = 1 10 r 3 a r a r - 1 2 is equal to

Step-by-step solution

Since, a r is the coefficient of x 10 - r in the binomial expansion of 1 + x 10 , therefore a r = C 10 - r 10 = C r 10 a r - 1 = C r - 1 10 So, a r a r - 1 = C r 10 C r - 1 10 ⇒ a r a r - 1 = 10 ! r ! × 10 - r ! × r - 1 ! × 11 - r ! 10 ! ⇒ a r a r - 1 = 11 - r r Now, ∑ r = 1 10 r 3 a r a r - 1 2 = ∑ r = 1 10 r 3 11 - r r 2 ⇒ ∑ r = 1 10 r 3 a r a r - 1 2 = ∑ r = 1 10 r 11 - r 2 ⇒ ∑ r = 1 10 r 3 a r a r - 1 2 = ∑ r = 1 10 r 121 + r 2 - 22 r &#