An agency entrusted to accredit colleges looks at four parameters: faculty quality $(F)$, reputation $(R)$. placement quality $(P)$, and infrastructure (I). The four parameters are used to arrive at an overall score, which the agency uses to give an accreditation to the colleges. In each parameter, there are five possible letter grades given, each carrying certain points:$A$ ($50$ points), $B$ ($40$ points), $C$ ($30$ points), $D$ ($20$ points), and $F$ ($0$ points). The overall score for a college is the weighted sum of the points scored in the four parameters. The weights of the parameters are $0.1$, $0.2$,$0.3$ and $0.4$ in some order, but the order is not disclosed. Accreditation is awarded based on the following scheme:

$\begin{array}{|cl|cl|}\hline

&\text{Range} && \text{Accreditation}\\ \hline

& \text{Overall score $\geq$ 45} & \text{AAA} \\ \hline

& \text{35 $\leq$ Overall score <45} & \text{BAA} \\ \hline

& \text{25 $\leq$ Overall score <35} & \text{BBA} \\ \hline

& \text{15 $\leq$ Overall score <25} & \text{BBB} \\ \hline

&\text{Overall score <15} & \text{Junk} \\ \hline \end{array}$

Eight colleges apply for accreditation, and receive the following grades in the four parameters ($F$, $R$, $P$, and $I$):

$\begin{array}{|cl|cI|cI|cI|}\hline

&\text{} & \text{F} & \text{R} & \text{P} & \text{I} \\ \hline

&\text{A-one} & \text{A} & \text{A} & \text{A} & \text{B} \\ \hline

&\text{Best Ed} & \text{B} & \text{C} & \text{D} & \text{D} \\ \hline

&\text{Cosmopolitan} & \text{B} & \text{D} & \text{D} & \text{C} \\ \hline

&\text{Dominance} & \text{D} & \text{D} & \text{B} & \text{C} \\ \hline

&\text{Education Aid} & \text{A} & \text{A} & \text{B} & \text{A} \\ \hline

&\text{Fancy} & \text{A} & \text{A} & \text{B} & \text{B} \\ \hline

&\text{Global} & \text{C} & \text{F} & \text{D} & \text{D} \\ \hline

&\text{High Q} & \text{C} & \text{D} & \text{D} & \text{B} \\ \hline \end{array}$

It is further known that in terms of overall scores:

- High $Q$ is better than Best $Ed$;
- Best $Ed$ is better than Cosmopolitan; and
- Education Aid is better than $A$-one.

How many colleges have overall scores between $31$ and $40$, both exclusive?

- $1$
- $3$
- $0$
- $2$