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 sum of all the $4$-digit numbers formed by taking all the digits from ${2, 3, 5, 7}$ without repetition is:

What is the total number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are the four cards of the same suit?

Let $P_m$ stand for $^nP_m$. Then the expression $1 \cdot P_1 + 2 \cdot P_2 + 3 \cdot P_3 + \dots + n \cdot P_n =$

$A$ five-digit number divisible by $3$ is to be formed using the digits $0, 1, 2, 3, 4, 5$ without repetition. The total number of ways this can be done is:

If all the letters of the word $COMBINATION$ are arranged in all possible ways to form $11$ letter words (with or without meaning),then the number of words among them in which $C$ and $N$ occupy the end positions and no vowel appears exactly in the middle position is