Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$ and $$10 \times 5,$$ respectively. The minimum number of scalar multiplications required to find the product $${A_1}{A_2}{A_3}{A_4}$$ using the basic matrix multiplication method is _________.

If the following system has non - trivial solution $$$px+qy+rz=0$$$ $$$qx+ry+pz=0$$$ $$$rx+py+qz=0$$$

Then which one of the following Options is TRUE?

Let $${a_n}$$ represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for $${a_n}$$?

$$\sum\limits_{x = 1}^{99} {{1 \over {x\left( {x + 1} \right)}}} $$ = _____________.