Permutation

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.

The number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently, is

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.

Show that $\frac{12!}{5!7!}+\frac{12!}{6!6!}=\frac{13!}{6!7!}$

Among the inequalities below, which ones are true for all natural numbers $n$ greater than $1000$ ?
I. $n!\le n^n$
II. ${\left(n!\right)}^2\le n^n$
III. ${10}^n\le n!$
IV. $n^n\le \left(2n\right)!$

The largest value of $n \in N$ for which $\frac{74}{{}^{n}P_n}>\frac{{n+3}_{P_3}}{n+1_{P_{n+1}}}$ is ________________

A code word of length $4$ consists of two distinct consonants in the English alphabet followed by two digits from $1$ to $9$, with repetition allowed in digits. If the number of code words, so formed ending with an even digit is $432k$, then $k$ is equal to

Suppose that six students, including Madhu and Puja, are having six beds arranged in a row. Further, suppose that Madhu does not want a bed adjacent to Puja. Then, the number of ways, the beds can be allotted to students are

The number of ways in which $3$ book can be placed in $4$ racks with at most one book in each rack is:

The number of ways in which a Six digit number can be formed such that even digits are in increasing order and odd digits are in decreasing order using $1,2,3,4,5,6$ (repetition is not allowed) is

In how many different ways can the letters of the word LEADING be arranged so that the vowels always come together?

The number of ways of arranging $8$ men and $4$ women around a circular table such that no two women can sit together, is

Ten different letters of an alphabet are given, words with five letters are formed from these given letters. Then, the number of words which have at least one letter repeated is

How many three-digit numbers can be formed without using the digits $0, 2, 3, 4, 5$ and $6$?

If $N$ denotes the number of different selections of 5 letters from the word $\mathbf{W}=$ MISSISSIPPI, then N belongs to the set:

The number of ways in which $3$ prizes can be distributed to $4$ children, so that no child gets all the three prizes, are

The set $A$ has $4$ elements and the set $B$ has $5$ elements$,$ then the number of injective mappings that can be defined from $A$ to $B$ is

How many numbers consisting of 5 digits can be formed in which the digits 3, 4 and 7 are used only once and the digit 5 is used twice?

What is the sum of the last two digits of the integer $1!+2!+3!+\dots +2005! ?$