Question

For any nonempty set $T$ whose elements are positive integers, define $f(T)$ to be the square of the product of the elements of $T$. For example, if $T=\{1,3,6\}$, then $f(T)=(1\cdot 3\cdot 6)^2 = 18^2 = 324$. Consider the nonempty subsets $T$ of $\{1,2,3,4,5,6,7\}$ that do not contain two consecutive integers. If we compute $f(T)$ for each such set, then add up the resulting values, what do we get?

Answer 1

We get

$225+324+441+576+784+1125+2304+3136+4900+11025=24840$

Explanation:

$(1\cdot 3\cdot 5)^2 = 15^2 = 225$

$(1\cdot 3\cdot 6)^2 = 18^2 = 324$

$(1\cdot 3\cdot 7)^2 = 21^2 = 441$

$(1\cdot 4\cdot 6)^2 = 24^2 = 576$

$(1\cdot 4\cdot 7)^2 = 28^2 = 784$

$(1\cdot 5\cdot 7)^2 = 35^2 = 1125$

$(2\cdot 4\cdot 6)^2 = 48^2 = 2304$

$(2\cdot 7\cdot 7)^2 = 56^2 = 3136$

$(2\cdot 5\cdot 7)^2 = 70^2 = 4900$

$(3\cdot 5\cdot 7)^2 = 105^2 = 11025$

We get

$225+324+441+576+784+1125+2304+3136+4900+11025=24840$