Problem $3.(10$ points$)$ Suppose you have a list of three numbers: $1,2,$ and $3.$ There are $36$ possible initial

arrangements: $A_1=[1,2,3],A_2=[1,3,2],A_3=[2,1,3],A_4=[2,3,1],A_5=[3,1,2],A_6=[3,2,1].$ We would like to

use the bubble sort algorithm to rearrange $($sort$)$ the elements in increasing order.

$($a$)(5$ points$)$ For each of the six arrangements, find the number of swaps produced by the bubble sort algorithm.

$($b$)(5$ points$)$ Find the average number of swaps produced by the bubble sort algorithm by computing the arithmetic mean

of the numbers you obtained in part $($a$).$