Permutations

Let $\mathbb N$ denote the set of all positive integers. Consider the sets
$$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$$
Let $S$ be the set of all functions $f:A\to B$ such that $f(2)\ne2$ and $f(4)\ne4$. Consider the set
$$T=\left\{f\in S:\text{there exists a function }g:B\to\mathbb N\text{ such that }g(f(x))=2^x\text{ for all }x\in A\right\}.$$
Then the number of elements in the set $T$ is ________.

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

Let $X$ be the set of all five digit numbers formed using 1, 2, 2, 2, 4, 4, 0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to _____.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is

Let $b_1b_2b_3b_4$ be a 4-element permutation with $b_i\in\{1,2,3,\dots,100\}$ for $1\le i\le4$ and $b_i\ne b_j$ for $i\ne j$, such that either $b_1,b_2,b_3$ are consecutive integers or $b_2,b_3,b_4$ are consecutive integers. Then the number of such permutations $b_1b_2b_3b_4$ is equal to ______.

The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ______.

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______.

$\displaystyle\sum_{r=1}^{20}(r^2+1)(r!)$ is equal to

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _______.

Let
$$\begin{aligned}S_1 &= \{(i, j, k) : i, j, k \in \{1, 2, \ldots, 10\}\},\\ S_2 &= \{(i, j) : 1 \le i < j + 2 \le 10,\ i, j \in \{1, 2, \ldots, 10\}\},\\ S_3 &= \{(i, j, k, l) : 1 \le i < j < k < l,\ i, j, k, l \in \{1, 2, \ldots, 10\}\}\end{aligned}$$
and
$$S_4 = \{(i, j, k, l) : i, j, k \text{ and } l \text{ are distinct elements in } \{1, 2, \ldots, 10\}\}.$$
If the total number of elements in the set $S_r$ is $n_r$, $r = 1, 2, 3, 4$, then which of the following statements is (are) TRUE?

Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x} =$

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other, is :

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is :

If the four letter words (need not be meaningful ) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is R and the fourth letter is E, then the total number of all such words is :

If $\dfrac{{}^{n+2}C_6}{{}^{n-2}P_2}=11$, then n satisfies the equation :