Permutations and Combinations

Five numbers $x_1,x_2,x_3,x_4,x_5$ are randomly selected from the numbers $1,2,3,\dots,18$ and are arranged in the increasing order $(x_1<x_2<x_3<x_4<x_5)$. The probability that $x_2=7$ and $x_4=11$ is :

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

Let A be a matrix of order $2\times2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p$, $2<p<8$, then the number of such matrices A is ______.

Let $S=\{E_1,E_2,\dots,E_8\}$ be a sample space of a random experiment such that $P(E_n)=\dfrac n{36}$ for every $n=1,2,\dots,8$. Then the number of elements in the set $\left\{A\subseteq S:P(A)\ge\dfrac45\right\}$ 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

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

The number of ways to distribute 30 identical candies among four children $C_1$, $C_2$, $C_3$ and $C_4$ so that $C_2$ receives atleast 4 and atmost 7 candies, $C_3$ receives atleast 2 and atmost 6 candies, is equal to :

Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f:S\times S\to S:f\text{ is onto and }f(a,b)=f(b,a)\ge a\ \forall(a,b)\in S\times S\}$ is ______.

If the constant term in the expansion of $\left(3x^3-2x^2+\dfrac5{x^5}\right)^{10}$ is $2^k\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to :

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 total number of four digit numbers such that each of first three digits is divisible by the last digit, 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 total number of functions, $f:\{1,2,3,4\}\to\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :

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 ______.

The number of bijective functions $f:\{1,3,5,7,\dots,99\}\to\{2,4,6,8,\dots,100\}$, such that $f(3)\ge f(9)\ge f(15)\ge f(21)\ge\dots\ge f(99)$, is ______

Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T\subseteq A:\text{ either }1\notin T\text{ or }2\in T\}$ and $C=\{T\subseteq A:T\text{ the sum of all the elements of }T\text{ is a prime number}\}$. Then the number of elements in the set $B\cup C$ is ______.

The number of 5-digit natural numbers, such that the product of their digits is 36, is ______.

Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{3,6,7,9\}$. Then the number of elements in the set $\{C\subseteq A:C\cap B\ne\phi\}$ 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 ______.

The number of matrices $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d\in\{-1,0,1,2,3,\ldots,10\}$, such that $A=A^{-1}$, is ______.