Let $$G\left( x \right) = 1/\left( {1 - x} \right)2 = \sum\limits_{i = 0}^\infty {g\left( i \right)\,{x^1}} \,\,\,,$$
where $$\left| x \right| < 1$$ What is $$g(i)$$?

Let $$n = {p^2}q,$$ where $$p$$ and $$q$$ are distinct prime numbers. How many numbers $$m$$ satisfy $$1 \le m \le n$$ and $$gcd\left( {m.n} \right) = 1?$$ Note that $$gcd(m,n)$$ is the greatest common divisor of $$m$$ and $$n$$.

The recurrence equation
$$\,\,\,\,\,\,\,T\left( 1 \right) = 1$$
$$\,\,\,\,\,\,T\left( n \right) = 2T\left( {n - 1} \right) + n,\,n \ge 2$$
evaluates to

In how many ways can we distribute 5 distinct balls, $${B_1},{B_2},......,{B_5}$$ in 5 distinct cells, $${C_1},{C_2},.....,{C_5}$$ such that Ball $${B_i}$$ is not in cell $${C_i}$$, $$\forall i = 1,2,....,5$$ and each cell contains exactly one ball?