Fundamental Principle of Counting

An archaeologist has $7$ coins of different value. What is the number of different sums of money the archaeologist can form?

In how many ways the word "PAPNAI" can be arranged.

Find the rank of the word "HATE" in the dictionary.

If ${}^{\mathrm{r}}{\mathrm{P}}_{\mathrm{r}-4}=30$ then, find the value of $\mathrm{r}$.

How many words can be made from "ROMANTIC", If three vowels are taken together.

Find the rank of the first word starting with K of the word "MONIKA". If all the letters are arranged according to the dictionary.

How many combinations can be formed by the word "INTESTINE" taking all word together.

If ${}^4{\mathrm{P}}_{\mathrm{r}}={\left(-30\right)}^{ 4}{\mathrm{P}}_{\mathrm{r}-1}$ then, find the value of $\mathrm{r}$.

If ${}^8{\mathrm{P}}_{\mathrm{r}}{:}^{ 8}{\mathrm{P}}_7=1:4!$ then, find the value of $\sqrt{\mathrm{r}}$.

$4$ women enter in a room together in which $7$ seats are available. Find the different ways they can seat.

In how many ways the word containing seven letters i.e. VBODMAS can be arranged?

$3$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? 

In how many different ways can the letters of the word PARTY be arranged?

$4$ boys and $2$ girls are to be seated in a row in such a way that the two girls are always together. In how many different ways can they be seated?

Three dice are rolled. The number of possible outcomes in which at least one dice shows$5$ is _______.

What is the probability of forming the word from the letters of the word $\text{'}IMPEACH\text{'}$ such that all vowels come together?

How many ways the letters of the $\mathrm{ASTIK}$ can be arranged, so that the vowels should not come together?

How many Numbers between $300$ and $1300$ can be formed using digits $0,1,2,3,4$ when repetition of digits are not allowed?