Permutations

All the 7 digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once and not divisible by $5$ are arranged in the increasing order. Find the ${\left(2004\right)}^{\mathrm{th}}$ number in this list.

How many $6$ digits odd numbers greater than $600000$ can be formed from the digits $5,6,7,8,9,0$ if

(a) repetitions are not allowed?

(b) repetitions are allowed?

Find the number of ways in which the number $30$ can be partitioned into three unequal parts, each part being a natural number. What this number would be if equal parts are also included.

Prove that: ${}^n{P}_r{=}^{ n-1}{P}_r+r{·}^{n-1}{P}_{r-1}$

Let the number of ways in which the letters of the word? 'MUNMUN' can be arranged so that no two alike letters are together are $N$, then value of $N$ is

If the sum of all numbers greater than $10000$ formed by using the digits $0,1,2,4,5$ & no digit being repeated in any number, is $N$, then sum of digits of $N$ is

Let $4$ digit numbers are there which contain not more than $2$ different digits. If total numbers are $N$, then $N$ is

If $P\left(n, n\right)$ denotes the number of permutations of $n$ different things taken all at a time then $P\left(n, n\right)$ is also identical to :-

All the $7$ digit numbers containing each of the digits $1, 2, 3, 4, 5, 6, 7$ exactly once and not divisible by $5$ are arranged in the increasing order. Then -

The number of five digit numbers that can be formed using all the digits $0, 1, 3, 6, 8$ which are -

Number of different words that can be formed using all the letters of the word "$\mathrm{DEEPMALA}$" if two vowels are together and the other two are also together but separated from the first two is -

The sum of all numbers greater than $1000$ formed by using the digits $1, 3, 5, 7$ such that no digit is being repeated in any number is -

The maximum number of different permutations of $4$ letters of the word "EARTHQUAKE" is -

Number of numbers greater than a million and divisible by $5$ which can be formed by using only the digits $1, 2, 1, 2, 0, 5$ & $2$ is -

The number of ways of arranging the letters $AAAAA, BBB, CCC, D, EE$ and $F$ in a row if no two ‘$C$’s are together :

Ten different letters of alphabet are given. Words with four letters are formed from the these letters, then the number of words which have at least one letter repeated is -

Boxes numbered $1, 2, 3, 4$ and $5$ are kept in a row and they are necessarily to be filled with either a red or a blue ball such that no two adjacent boxes can be filled with blue balls. How many different arrangements are possible, given that the balls of a given colour are exactly identical in all respects?

The number of ways in which $5$ different books can be distributed among $10$ people if each person can get at most one book is -

Out of seven consonants and four vowels, the number of words of six letters, formed by taking four consonants and two vowels is (Assume that each ordered group of letter is a word):

The number of different seven digit numbers that can be written using only three digits $1, 2$ & $3$ under the condition that the digit $2$ occurs exactly twice in each number is -