Let $$f\left( {w,x,y,z} \right) = \sum {\left( {0,4,5,7,8,9,13,15} \right).} $$ Which of the following expressions are NOT equivalent to $$f?$$
$$(P)\,\,\,$$ $$x'y'z' + w'xy' + wy'z + xz$$
$$(Q)\,\,\,$$ $$w'y'z' + wx'y' + xz$$
$$(R)\,\,\,$$ $$w'y'z' + wx'y' + xyz + xy'z$$
$$(S)\,\,\,$$ $$x'y'z' + wx'y' + w'y$$

Consider the following Boolean function with four variables
$$F\left( {w,\,x,\,y,\,z} \right) = \sum {\left( {1,\,3,\,4,\,6,\,9,\,11,\,12,\,14} \right)} $$ the function is

The maximum number of binary trees that can be formed with three unlabeled nodes is:

Let $$A$$ be the matrix $$\left[ {\matrix{ 3 & 1 \cr 1 & 2 \cr } } \right]$$. What is the maximum value of $${x^T}Ax$$ where the maximum is taken over all $$x$$ that are the unit eigenvectors of $$A$$?