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 total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$,find $x$.

$A$ number is called a palindrome if it reads the same backward as well as forward. For example,$285582$ is a six-digit palindrome. The number of six-digit palindromes,which are divisible by $55$,is ...... .

The number of numbers between $2000$ and $5000$ that can be formed with the digits $0, 1, 2, 3, 4$ (repetition of digits not allowed) and are multiples of $3$ is:

The remainder obtained when $1! + 2! + 95!$ is divided by $15$ is