AMC8 2018

Answer (C): Think of the $+$ sign as $+1$, and the $-$ sign as $-1$. Let $a$, $b$, $c$, and $d$ denote the values of the four cells at the bottom of the pyramid, in that order. Then the cells in the second row from the bottom have values $a\cdot b$, $b\cdot c$ and $c\cdot d$, and the cells in the row above this are $a\cdot b\cdot b\cdot c=a\cdot c$ and $b\cdot c\cdot c\cdot d=b\cdot d$ (because both $1$ and $-1$ squared are $1$). Finally, the top cell has value $a\cdot b\cdot c\cdot d$. This value is $+1$ if all four variables are $+1$ or all four are $-1$, giving two ways; or, if two of the variables are $+1$ and two are $-1$, giving $6$ additional ways ($++--$, $+-+-$, $+--+$, $-++-$, $-+-+$, and $--++$). Thus there are a total of $8$ ways to fill the fourth row.