Given a set of coins with values $(V_1,V_2,dots{V}_N)$ and a target sum $S,$ find the fewest coins required to equal $S.$ To find minimum number of coins to sum to $15$ with values $\{1,5,10\},$ Let $M(S)=$ minimum number of coins to sum to $S,$ start with sum $0,$ using dynamic programming approach to find optimized solutions at each step.$(6)$