Permutations

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.

Let the digits $a, b, c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is$\_\_\_\_\_\_\_.$

A person forgets his $4$-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is________.

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to ____.

The total number of three-digit numbers, divisible by $3$, which can be formed using the digits $1, 3, 5, 8$, if repetition of digits is allowed, is 

The number of five-digit numbers, greater than $40000$ and divisible by $5$, which can be formed using the digits $0, 1, 3, 5, 7$ and $9$ without repetition, is equal to

The number of permutations, of the digits $1, 2, 3, \dots , 7$ without repetition, which neither contain the string $153$ nor the string $2467$, is _______ .

The number of $4$-letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which $C$ and $S$ do not come together, is $(6!)k$ then $k$ is equal to

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

How many three-digit number can be formed which are divisible by $3$ using the digits $1,3,5,8$ and repetition is allowed ?

The rank of the word "MONDAY" is 

The number of seven digit numbers using $1,2,3,4$ whose sum of digits is $12$ is