Factorial Notation

Which of the following numbers will completely divide$(3^{19} + 3^{20} + 3^{21} + 3^{22}) ?$

A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1's and 2's. Find the total number of ways is in which the person can go up the stairs.

Eighteen guests have to be seated half on each side of a long table. Four particular guests desire to sit on one particular side and three other on the other side. Determine the number of ways in which the sitting arrangements can be made.

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

The number of five-digit numbers, greater than $40000$ and divisible by $5$, which can be formed using the digits $0, 1, 3, 5, 7$ and $9$ without repetition, is equal to

The number of permutations, of the digits $1, 2, 3, \dots , 7$ without repetition, which neither contain the string $153$ nor the string $2467$, is _______ .

The number of $4$-letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

The rank of the word "MONDAY" is 

Using all the letters of the word "MATHS", the rank of the word "THAMS" is 

The rank of the word "PUBLIC" is 

In how many ways can $3$ boys and $4$ girls be arranged in a row that all three boys are together?

How many numbers between $400$ to $1000$ (both exclusive) can be made with digits $2,3,4,5,6,0$ if repetition of digits is allowed.

If ${\left(10!\right)}^2-4\times {\left(8!\right)}^2$ is divisible by $2^n$, then what is the greatest value of $n$?

Compute: $\frac{12!}{\left(10!\right)\left(2!\right)}$

Evaluate: $7!-5!$

Evaluate: $7!$

$\frac{1}{2!}+\frac{2}{3!}+\frac{{\displaystyle 3}}{{\displaystyle 4!}}+.......+\frac{{\displaystyle 9}}{{\displaystyle 10!}}=1-\frac{{\displaystyle 1}}{{\displaystyle 10!}}$

For $k\in N$, let $\frac{1}{\alpha \left(\alpha +1\right)\left(\alpha +2\right)\left(\alpha +3\right)...\left(\alpha +20\right)}=\sum _{k=0}^{20}\frac{A_k}{\alpha +k}$ where $\alpha >0$, then the value of $100{\left(\frac{A_{14}+A_{15}}{A_{13}}\right)}^2$ is