Consider the following system of equations:
3x + 2y = 1
4x + 7z = 1
x + y + z =3
x - 2y + 7z = 0
The number of solutions for this system is ______________________

Consider an undirectional graph $$G$$ where self-loops are not allowed. The vertex set of $$G$$ is $$\left\{ {\left( {i,j} \right):\,1 \le i \le 12,\,1 \le j \le 12} \right\}.$$ There is an edge between $$(a,b)$$ and $$(c,d)$$ if $$\left| {a - c} \right| \le 1$$ and $$\left| {b - d} \right| \le 1$$. The number of edges in this graph is _____.

There are 5 bags labeled 1 to 5. All the coins in given bag have the same weight. Some bags have coins of weight 10 gm, other have coins of weight 11 gm. $${\rm I}$$ pick 1, 2, 4, 8, 16 coins respectively from bags 1 to 5. Their total weight comes out to 323 gm. Then the product of the labels of the bags having 11 gm coin is _______ .

The maximum number of edges in a bipartite graph on $$12$$ vertices is _________.