$1.$ The $10-$character string "STATUESQUE" contains $$S$2$ times, $$T$2$ times, $$U$2$ times

and $$E$2$ times. The remaining $2$ characters occur $1$ time each.

$($a$)$ How many $10-$character strings are arrangements of this string?

$($b$)$ How many of the strings in part $($a$)$ contain none of the substrings $"$SS$","$TT$","$UU$","$EE$"?$

$($c$)$ How many of the strings in part $($a$)$ contain at least one of the four substrings in

part $($b$)?$

$($d$)$ How many of the strings in part $($a$)$ contain exactly one of the four substrings in

part $($b$)?$