Not sure what your solution is or why are you taking max but i guess the solution should look something like this :

$$$dp[val][cnt] = \displaystyle\sum_{x=1}^{val} dp[val-x][cnt-1]$$$ with base case $$$dp[0][0] = 1$$$

Where $$$dp[i][j]$$$ means the number of ways to get sum $$$i$$$ with exactly $$$j$$$ coins. You can answer all the other types using these dp values.