Permutations when All the Objects are Distinct

Let $X$  be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to$Y$ and $\beta$ is the number of onto functions from $Y$ to  $X$ , then the value of $\frac{1}{5!}\left(\beta -\alpha \right)$ is

A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1's and 2's. Find the total number of ways is in which the person can go up the stairs.

Eighteen guests have to be seated half on each side of a long table. Four particular guests desire to sit on one particular side and three other on the other side. Determine the number of ways in which the sitting arrangements can be made.

Five speakers $A, B, C, D$ and $E$ are to be scheduled to speak such that $A$ and $B$ must speak after one another. In how many ways can their speeches be scheduled?

In a unique hockey series between India and Pakistan, they decide to play on till a team wins $5$ matches. The number of ways in which the series can be won by India, if no match ends in a draw, is

The number of permutations of the letters $a_1,a_2,a_3,a_4,a_5$ in which the first letter $a_1$ does not occupy the first position (from the left) and the second letter $a_2$ does not occupy the second position (from the left) is

How many $3-$digit numbers can be formed by using the digits $1$ to $9$, if  no digit is repeated?

Six candidates are called for interview to fill some vacant posts in an office. Assuming that each candidate is fit for each post, determine the number of ways in which four posts can be filled.

How many $3-$digit numbers can be formed from the digits $1, 2, 3, 4$ and $5$, if repetition of digits is not allowed?

Find the value of $P(4, 3)$.

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

The number of $5$-letter words that can be formed by using the letters of the word $\mathrm{SARANAM}$ is

If $x$ is the number of ways in which six women and six men can be arranged to sit in a row such that no two women are together and if $y$ is the number of ways they are seated around a table in the same manner, then $x:y=$

The sum of four digit even numbers that can be formed with the digits $0, 3, 5, 4$ without repetition is

The number of ways of arranging $8$ boys and $8$ girls in a row so that boys and girls sit alternately is

If ${}^{12}{P}_r{=}^{11}{P}_6+6{\times }^{11}{P}_5$, then $r$ 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? (Express your answer as only numerical value)

How many $6$-digit numbers can be formed from the digits $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated?