Permutations

If the set $A$ contains $7$ elements and the set $B$ contains $10$ elements, then the number of one-one functions from $A$ to $B$ is

In how many ways can $6$ boys be arranged in a row?

In how many ways can $10$ books be arranged in a shelf so that a particular pair of books shall be never together?

In how many ways can $10$ books be arranged in a shelf so that a particular pair of books shall be always together?

In how many ways can the letters of the word 'PENCIL' be arranged so that N is always next to E?

In how many ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only odd positions?

How many words can be formed from the letters of the word 'LAUGHTER' so that the vowels are never together?

How many words can be formed from the letters of the word 'DAUGHTER'  so that the vowels always come together?

How many words beginning with T and ending with E can be made with no letters repeated out of the letters of the word 'TRIANGLE'?

The number of positive integers greater than $6000$ and less than $7000$ which are divisible by $5$ with no digit repeated, is

How many $3$-digit even numbers can be formed with no digit repeated by using the digits $0,1,2,3,4$ and $5$?

Three different rings are to be worn in $4$ fingers. In how many ways can this be done?

Ten students are participating in a race. In how many different ways can the first prize be won?

In how many number of ways, the letters of the word 'OPTICAL' can be arranged so that the vowels always come together?

If $P_r$ stands for ${}^r{P}_r$ then the sum of the series $1+P_1+2P_2+3P_3+\dots +n{P}_n$ is

The number of $10$ digit number that can be written by using the digits $2$ and $3$ is

The sum of the digits at the ${10}^{th}$ place of all numbers formed with the help of $2, 4, 5, 7$ taken all at a time is

In an examination there are three multiple choice questions and each question has $5$ choices. Number of ways in which a student can fail to get all answer correct is

The number of five digit telephone numbers having at least one of their digits repeated is

The product of $r$ consecutive positive integers is divisible by