Exercise $1.$ Hopping Up Steps $-$ recursive formula

We are given the following problem: figure out how many ways we can hop up n steps, if at every point we can make one hop of d steps, where d is one of the following $\{d1,d2,$dotsdk$\}?$

For example, if we can hop either one or two steps at a time, there are FIVE valid ways to hop $4$ steps: $\{1,1,1,1\},\{2,1,1\},\{1,2,1\},\{1,1,2\},\{2,2\}.$

This problem is somewhat similar to the problem of finding out the minimum number of coins to make change for amount n with denominations $\{d1,d2,$dotsdk$\}.$

Note: When k is $1($there is only one type of hop to make$),$ the answer should not be more than $1.$

a$)$ Find the recursive formula for computing findNumWays$($n$,\{d1,d2,$dotsdk$\}).$

Carefully explain why your formula is correct.

b$)$ Create working Python code for findNumWaysR, which is a recursive method based on your formula in $($a$),$ and test that it works correctly. Test it for various inputs, such as findNumWaysR $(4,\{1,2\})$ findNumWaysR$(5,\{17\},$findNumWaysR $\{17,\{1,3,5\}$