Question 461595
First up, let S = (1+a)(1 + b)(1 + c)(1 + d) = 1+c + d + cd + b + bc + bd + bcd + a + ac + ad + acd + ab + abc + abd + abcd.  (This expression has 16 terms.)
By the AM-GM inequality, we have

{{{S/16 >= root(16, cdcdbbcbdbcdaacadacdababcabdabcd) = root(16, a^8b^8c^8d^8) = root(16, (abcd)^8) = root(16,1) = 1}}},  since abcd = 1.

Hence {{{S >= 16 = 2^4}}}