The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

If the number of seven-digit numbers,such that the sum of their digits is even,is $m \cdot n \cdot 10^{n}$; $m, n \in \{1, 2, 3, \ldots, 9\}$,then $m+n$ is equal to . . . . . . .

$2^n \{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n - 3) \cdot (2n - 1)\} = \dots$

If $S_n = \sum_{r=0}^n \frac{1}{^nC_r}$ and $T_n = \sum_{r=0}^n \frac{r}{^nC_r}$,then $\frac{T_n}{S_n}$ is equal to

The number of words that do not start and end with vowels,formed using all the letters of the word $'UNIVERSITY'$ such that all vowels are in alphabetical order,is

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90(X)$,then $X=$