$1.$ Let $\backslash ($ A $\backslash )$ be the set of all strings of decimal digits of length $6.$ For example $241576$ and $051891$ are strings in $\backslash ($ A $\backslash ).$

$($a$)$ How many strings in $\backslash ($ A $\backslash )$ have exactly two $5\text{'}$s$?$

$($b$)$ How many strings in $\backslash ($ A $\backslash )$ have at least two $6\text{'}$s$?$

$($c$)$ How many strings in $\backslash ($ A $\backslash )$ have the digits in a strictly increasing order? For example $1234678$ and $2345678$ are such strings, but $202841$ and $123557$ are not.