2014 KIMC keystage3-IWYMIC team

Let $S_1-S_2+S_3-S_4+S_5=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and

$$\begin{aligned}S_1&=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}.\end{aligned}$$

$$\begin{aligned}S_2&=\frac{1}{2\times3}+\frac{1}{2\times4}+\frac{1}{2\times5}+\frac{1}{2\times6}+\frac{1}{3\times4}\\ &\quad+\frac{1}{3\times5}+\frac{1}{3\times6}+\frac{1}{4\times5}+\frac{1}{4\times6}+\frac{1}{5\times6}.\end{aligned}$$

$$\begin{aligned}S_3&=\frac{1}{2\times3\times4}+\frac{1}{2\times3\times5}+\frac{1}{2\times3\times6}+\frac{1}{2\times4\times5}+\frac{1}{2\times4\times6}\\ &\quad+\frac{1}{2\times5\times6}+\frac{1}{3\times4\times5}+\frac{1}{3\times4\times6}+\frac{1}{3\times5\times6}+\frac{1}{4\times5\times6}.\end{aligned}$$

$$\begin{aligned}S_4&=\frac{1}{2\times3\times4\times5}+\frac{1}{2\times3\times4\times6}+\frac{1}{2\times3\times5\times6}+\frac{1}{2\times4\times5\times6}+\frac{1}{3\times4\times5\times6}.\end{aligned}$$

$$\begin{aligned}S_5&=\frac{1}{2\times3\times4\times5\times6}.\end{aligned}$$

Determine the value of $m+n$.

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