consider the uniform distribution on the set of formulas. Note that the formulas we consider are $2-$cores. (Here the same equation to occur several times. This happens with probability $o(1)$ as $m$ is linear in $n$ and can be ignored. ) Let $X$ be the number of solutions of a formula. We have $X=\sum_a\, X_a$ where $a$ stands for an assignment of the variables with $0,1, 2$ and $X_a(F)=1$ if $F$ is true under $a$ and $0$ otherwise. The expectation of $X$ is $3^{n-m}$ because given an assignment each equation is true independently with probability $1/3.$ We assume that $m=\gamma n , \gamma $ bounded above by a constant $<1.$ As $k$ is also constant, the asymptotics is only with respect to $n.$ We need to show the following theorem

\[EX2\] $\mbox{E}[X^2] \le  C \cdot 3^{2(n-m)} $

We have E$[X^2] \,= \, \sum_{(a, b)}\, \mbox{E}[X_a \cdot X_b] $ where $(a, b)$ refers to all ordered pairs of assignments.

Let $\overline{W}=(W_0, W_1, W_2)$ be a partition of the set of variables into $3$ sets.We always use the notation $w_i = \sharp W_i,\, \,  \bar{w}\, = \, (w_0, w_1, w_2).$ For two assignments we write $b \, = \, D(a, \overline{W}) $ 