Permutations

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____ .

If ${}^{2n+1}P_{n-1}:{}^{2n-1}P_n=11:21$, then $n^2+n+15$ is equal to :

Let $5$ digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is _____ .

The number of seven digits odd numbers, that can be formed using all the seven digits $1, 2, 2, 2, 3, 3, 5$ is

The number of square matrices of order $5$ with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is 

The number of $9$ digit numbers, that can be formed using all the digits of the number $123412341$ so that the even digits occupy only even places, is ______

The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is

The number of $5$-digit natural numbers, such that the product of their digits is $36$, is

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

Let $A$ be a $3\times 3$ matrix having entries from the set $\left\{-1,0,1\right\}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is _____

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

The total number of $5$-digit numbers, formed by using the digits $1,2,3,5,6,7$ without repetition, which are multiple of $6$, is

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is

A number is called a palindrome if it reads the same backward as well as forward. For example $285582$ is a six digit palindrome. The number of six digit palindromes, which are divisible by $55,$ is ________.

The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is______.

If ${}^{1}P_1+2·{}^{2}P_2+3·{}^{3}P_3+\dots +15·{}^{15}P_{15}={}^{q}P_r-s, 0\le s\le 1$, then ${}^{q+s}C_{r-s}$ is equal to

All the arrangements, with or without meaning, of the word $\mathrm{FARMER}$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $\mathrm{FARMER}$ in this list is _____ .

The sum of all the $4$-digit distinct numbers that can be formed with the digits $1, 2, 2$ and $3$ is:

The total number of $3\times 3$ matrices $A$ having enteries from the set $\left(0,1,2,3\right)$ such that the sum of all the diagonal entries of $A{A}^T$ is $9$, is equal to

The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1,2$ and $3$ only is: