Fundamental Principle of Counting

Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1, 2, 3, 4, 5$ with repetition, is ________

The largest natural number $n$ such that $3^n$ divides $66!$ is $\_\_\_\_\_\_\_$

The number of six digit number formed by using the digits $\left\{1,2,3,4,5,6\right\}$ which are divisible by $6$ (repetition is not allowed)

Let the number of matrices of order $3\times 3$ are possible using the digits $\left\{0,2,3,...,10\right\}$ is $m^n$, then $m+n$ is

$N>40000$, where $N$ is divisible by $5$. How many such $5$ digit numbers can be formed using $0,1,3,5,7,9$ without repition.

Using the number $1,2,3,....,7$, total numbers of $7$ digit number which does not contain string $154$ or $2367$ is, (repetition is not allowed)

Maximum value of $n$ such that $66!$ is divisible by $3^n$ is

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

Show that if $6$ integers are selected from first $10$ positive integers, there must be a pair of these integers with sum $11$.

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

If $n$ is a factor of $72$ such that $xy=n$, then number of ordered pairs $\left(x,y\right)$ where $x,y\in N$ is

If number of 5-digit number of the form $abcde$ where, $a,b,c,d,e\in \left\{0,1,2,...,9\right\}$ and $b=a+c$, $d=c+e$ is $\lambda$, then $\lambda$ is

The number of zero's at the end of $\left(2020\right)!$ is $N$ then $N-450$ is

The number of $3$ digit odd numbers, that can be formed by using the digits $1,2,3,4,5,6$ when the repetition is allowed, is

The number of ways in which $7$ pencils, $6$ books and $5$ pens be disposed off is

The number of quadratic expression with the coefficient drawn from set $\left\{0,1,2,3\right\}$ is 

The number of ways in which one can draw money form a bag containing five rupee coin, two rupee coin, one rupee coin, fifty paisa coin and twenty five paisa coin are

$n$ dice are rolled. The number of possible outcomes for which at least one of the dice shows an even number is $189,$then $n=$

Let $m$ be the number of values of $n\left(1\le n\le 2014\right)$ for which $\frac{8n}{999-n}$ is an integer. Find $10m$.