If a + b + c + d = 2, then the maximum value of (1 + a)(1 + b)(1 + c)(1 + d) is:

$$\eqalign{ & a + b + c + d = 2 \cr & {\text{Let }}a = b = c = d = \frac{1}{2} \cr & {\text{satisfy the above}} \cr & {\text{so, }}\left( {1 + a} \right)\left( {1 + b} \right)\left( {1 + c} \right)\left( {1 + d} \right) \cr & = \left( {1 + \frac{1}{2}} \right)\left( {1 + \frac{1}{2}} \right)\left( {1 + \frac{1}{2}} \right)\left( {1 + \frac{1}{2}} \right) \cr & = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \cr & = \frac{{81}}{{16}} \cr} $$