$2.$ A junior METU$-$IE student has final examinations in $3$ courses, $361,323,$ and $333,$ each worth the same number of credits. There are only $3$ days available for studying. Assume that the student has to devote a nonnegative integer number of the available days for studying for each course, i$.$e$.,$ a day cannot be split between two courses. Estimates of expected grades upon various numbers of days devoted for studying for each course are given below:

Table $1$: Expected grade based on number of days studied

a$.$ Determine the number of available days that the student should devote to each course in order to maximize the sum of all grades, using DP$.$make sure you formulated as dp modelling

b$.$ How does the strategy in $($a$)$ change if the student has $4$ days available to study before the examinations? What is the increase in the optimum objective value?

c$.$ How do the strategies in $($a$),($b$)$ change if a new course $-$elective$-$ is added, with expected grades of $\backslash (0,0,2,3\backslash ),$ and $4$ when the number of days devoted to studying it is $\backslash (0,1,2,3\backslash ),$ and $4$ respectively?