Fundamental Principle of Counting

A natural number $n$ such that $n!$ ends in exactly $1000$  zeros is

The number of distinct positive integers can be formed using $0, 1, 2, 3$ where each integer used at most once is equal to

The number of ways in which $\text{'}n\text{'}$ distinct objects can be put into two different boxes is

The number of points, at which the two curves $y=\frac{x}{99}$ and $y=\mathrm{sin\pi }x$ intersect, is

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated, is

4 buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, then the total possible ways are (bus can't take same path)

There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back, is

Find the number of arrangements of the letters of the word ASSASSINATION.

In how many different ways can the letters of the word "LOGITECH" be arranged in such a way that the vowels always come together?

Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is

Four normal dice are rolled once. The number of possible outcomes in which at least one die shows up $2$ is -

Find the number of arrangements of the letters of the word ASSASSINATION.

There are $5$ true / false questions in an examination. The possible number of sequences of answer is 

The number of $3-$digit number that end with $5$ is

In a class room there are $3$ entrances and two exits. In how many ways a student can enter into the room and then come out ?

${}^5{\mathrm{p}}_{\mathrm{r}}={}^6\mathrm{p}_{\mathrm{r}-1}$ $,$then $\mathrm{r}=$

If $\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}$, then $x$ is equal to

$3$girls and $4$ boys are to be seated in a row on $7$ chairs in such a way that all the three girls always sit together. In how many different ways, can it be done?