Question

find the coefficient of x^29 in the expansion (1+2x)^12 .(1+x)^18

Answer 1

(2x+1)^12
= (2x)^12 + 12(2x)^11 + ...
= 2^12 x^12 + 12*2^11 x^11 + ...
(x+1)^18 = x^18 + 18x^17 + 9*17 x^16 + ...
= 2^12 x^30 + (18*2^12 + 12)x^29 + ...

so, 18*2^12 + 12 = 73740

Answer 2

(2x+1)^12 = 2^12 x^12 + 12*2^11 x^11 + ...
(x+1)^18 = x^18 + 18x^17 + ...
multiply the two expansions to get

2^12 x^30 + (18*2^12 + 12*2^11) x^29 + ...

so, we have 18*2^12 + 12*2^11 = 98304