The value of ∑ r = 0 6 C r 6 · C 6 - r 6 is equal to :

Step-by-step solution

We have ∑ r = 0 6 C r 6 · C 6 - r 6 = C 0 6 · C 6 6 + C 1 6 · C 5 6 + . . . + C 6 6 · C 0 6 Now, we know that the expansion of 1 + x n = C 0 n x 0 + C 1 n x 1 + C 2 n x 2 + . . . C n n x n ⇒ ( 1 + x ) 6 ( 1 + x ) 6 = C 0 6 + C 1 6 x + C 2 6 x 2 + . . . + C 6 6 x 6 C 0 6 + C 1 6 x + C 2 6 x 2 + . . . + C 6 6 x 6 ⇒ ( 1 + x ) 12 = C 0 6 + C 1 6 x + C 2 6 x 2 + . . . + C 6 6 x 6 C 0 6 + C 1 6 x + C 2 6 x 2 + . . . + C 6 6 x 6 On comparing the coefficient of x 6 both sides, we get C