$1)$Create a network with exactly $100$ different paths from node $1$ to node n$.$ You can choose yourself what n is and which arcs are in the network.

Number of paths is $100.$

Write the python algorithm that creates the net:

$2)$Given a directed acyclic network with python, your task is to count how many different paths there are in the network.

Implement a class AllPaths with the following methods:

add$\_$edge adds an arc between two nodes

count gives the number of different paths

Implement method count efficiently with dynamic programming.

Implement the class in the file allpaths.py as shown in the following example.

class AllPaths:

def $\_\_$init$\_\_($self$,$ n$)$:

# TRUE

def add$\_$edge$($self$,$ a$,$ b$)$:

# TRUE

def count$($self$)$:

# TRUE

if $\_\_$name$\_\_=="\_\_$main$\_\_"$:

a $=$ AllPaths$(4)$

a$.$add$\_$edge$(1,2)$

a$.$add$\_$edge$(1,3)$

a$.$add$\_$edge$(2,4)$

a$.$add$\_$edge$(3,4)$

print$($a$.$count$())$ # $10$

a$.$add$\_$edge$(2,3)$

print$($a$.$count$())$ # $14$

Explanation: The first call of the count method counts the following paths:

$1$

$1->2$

$1->2->4$

$1->3$

$1->3->4$

$2$

$2->4$

$3$

$3->4$

$4$