$6\mathrm{.}3.$ Counting paths

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Before attempting the problems below read the section Networks and paths which explains how to compute the number of paths of a give

length in a network.

$6\mathrm{.}3.$a$.$ Counting paths $1$

$\frac{0\mathrm{.}0}{10\mathrm{.}0}$ points $($graded$)$

Consider the network shown in the image. A double headed arrow iharrj indicates

that there is an edge $ij$ and an edge $ji.$

Compute the number of paths in this network consisting of exactly $5$ edges, which start

at the vertex $1$ and end at the vertex $2.$

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