Factorial Notation

Show that $\frac{12!}{5!7!}+\frac{12!}{6!6!}=\frac{13!}{6!7!}$

Let number of values of '$r$' for which ${}^{15}C_r$ is an odd integer, be '$k$', then $k$ is greater than

If $a_n=\frac{n}{\left(n+1\right)!}$, then find $\sum _{n=1}^{50}{a}_n$.

Prove that the greatest value of ${}^{2n}\mathrm{C}_r$ is twice the greatest value of ${}^{2n-1}\mathrm{C}_r$.

If the permutations of the word MITUL are arranged in dictionary order, find the rank of the word MITUL.

All possible permutations of the word LAND are arranged in dictionary order. Find the rank of the word LAND.

If for two integers $\mathrm{n}$ and $\mathrm{m}(<\mathrm{n}),$$\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)...(\mathrm{n}-\mathrm{m})\mathrm{x}=\mathrm{n}! .$Then find $\mathrm{x}.$

If $(\mathrm{n}+2)!=2550. \mathrm{n}!,$then find $\mathrm{n}.$

If $\frac{1}{4!}+\frac{1}{5!}=\frac{\mathrm{x}}{6!},$then find $\mathrm{x}.$

Find the LCM of $6!, 8!,9!,11!.$

Find the LCM of $3!, 6!$ and $8!.$

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

Prove that for $n\ge 4, 1!+2!+\dots +n!$ cannot be the square of a positive integer,

How many different permutations can be form with the letters of the word 'DELHI'

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

The product of any $r$ consecutive natural numbers is always divisible by $r !$

Evaluate: $\frac{12!}{10!2!}$

Find the remainder when $1! +2!+3! +4!+.....+n!$ is divided by $15, if n\ge 5.$

The remainder obtained, when $1!+2!+3!+\dots +175!$, is divided by $15$ is $\lambda$ then value of $\frac{{\lambda }^3}{2}$ is

If $S=1+1·\mathrm{1!}+2·\mathrm{2!}+3·\mathrm{3!}+...........n·n!$, then the value of $S$ is equal to