The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is

The sixth term of an $A.P.$ is equal to $2$, the value of the common difference of the $A.P.$ which makes the product ${a_1}{a_4}{a_5}$ least is given by

In an $A.P.,$ if $p^{\text {th }}$ term is $\frac{1}{q}$ and $q^{\text {th }}$ term is $\frac{1}{p},$ prove that the sum of first $p q$ terms is $\frac{1}{2}(p q+1),$ where $p \neq q$

In $\Delta ABC$, if $a, b, c$ are in $A.P.$ (with usual notations), identify the incorrect statements -

Let $S_n$ denote the sum of the first $n$ terms of an $A.P$.. If $S_4 = 16$ and $S_6 = -48$, then $S_{10}$ is equal to

If $a,\;b,\;c,\;d,\;e,\;f$ are in $A.P.$, then the value of $e - c$ will be