Use the binomial theorem to find the 12th term in the binomi

Use the binomial theorem to find the 12th term in the binomial expansion of (-2x+1/2y)^19.

Find the 8th number in the 11th row (n=11) of Pascal\'s triangle.

Show that the sum of the 34th and the 35th numbers in the 117th row of Pascal\'s triangle is equal to the 85th number in the 118th row of the triangle

Solution

the 12th term in the expansion of   (-2x+1/2y)^19.

so C(n, k) x^ n-k y^k

we need the 12th term so plug k =11

C(19, 11) [-2 sqrt(x)]^8 . (1/2y)^11

the 12th rem = 19C11 2^8 x^4 . 2^-11 . y^-11

= 19C11 2^-3 x^4 . y^-11

the 8th number in the 11th row (n=11) of Pascal\'s triangle is 330