Permutations

Consider all the permutation of the word JANLOKPAL number of words which O never lies between J and N.

During school function for a particular program 4 boys and 3 girls are to be seated in a row wearing colourful dresses. Concerned teachers-in-charges are trying all possible ways of arrangements. Help them answering the following. (i) In how many ways can seated?

Find the value of $n$ such that $\frac{{}^{n}P_4}{{}^{n-1}P_4}=\frac{5}{3}, n>4$

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

Derivation of ${}^n{P}_r$

$5$ boys $\left(B_1,B_2,B_3,B_4 \text{and} B_5\right)$ and $5$ girls $\left(G_1,G_2,G_3,G_4 \text{and} G_5\right)$ are to be seated around a round table such that boy and girl sit alternately and $B_1$ does not sit beside $G_i\forall i\in \left\{1,2,3,4,5\right\}.$ If the number of such arrangements is $N$, then the sum of digits of $N$ is equal to $?$

In how many ways can $5$ girls and $3$ boys be seated in a row so that no two boys are together?

How many numbers greater than $1000000$ can be formed by using the digits $1, 2, 0, 2, 4, 2, 4$?

Find the number of words with or without meaning which can be made using all the letters of the word $\mathrm{AGAIN}$. If these words are written as in a dictionary, what will be the ${50}^{\mathrm{th}}$ word?

In how many ways can $4$ red, $3$ yellow and $2$ green discs be arranged in a row if the discs of the same colour are indistinguishable ?

Find the number of different $8$-letter arrangements that can be made from the letters of the word $\mathrm{DAUGHTER}$ so that all vowels occur together, and all vowels do not occur together.

Find $r$, if $5 {}^{4}P_r=6 {}^{5}P_{r-1}$

Find the value of $n$ such that ${}^{n}P_5=42{}^{n}P_3, n>4$

How many numbers lying between $100$ and $1000$ can be formed with the digits $0, 1, 2, 3, 4, 5$, if the repetition of the digits is not allowed?

How many $4$-digit numbers can be formed by using the digits $1$ to $9$ if repetition of digits is not allowed?

Find the number of permutations of the letters of the word $\mathrm{ALLAHABAD}$.

If $\frac{1}{8!}+\frac{1}{9!}=\frac{x}{10!}$, find $x$.

Evaluate $\frac{n!}{r!\left(n-r\right)!}$, when $n=5, r=2$.

Evaluate: $5!$

The total number of $6$ digited numbers greater than $400000$ that can be formed by using the digits $0, 2, 2, 4, 4, 5$ is