Fundamental Principle of Counting

Let $A$ is set of all possible planes passing through four vertices of given cube. Find number of ways of selecting four planes from set $A$, which are linearly dependent and one common point. (If planes $P_1=0$, $P_2=0,P_3=0$ and$P_4=0$ can be written as $a{P}_1+b{P}_2+c{P}_3+d{P}_4=0$, where all $a, b, c, d$ are not equal to zero, then we say planes $P_1,P_2,P_3,P_4$ are linearly dependent planes).

Prove that $\left(n!\right)!$ is divisible by $n{!}^{\left(n-1\right)!}$

Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.

Find the number of $4$-digit numbers (in base $10$ ) having non-zero digits and which are divisible by $4$ but not by $8$.

A user of facebook which is two or more days older can send a friend request to someone to join facebook. If initially there is one user on day one then find a recurrence relation for $a_n$ where $a_n$ is a number of users after $n$ days.

In a row, there are $81$ rooms, whose door no. are $1, 2 ,\dots \dots ., 81,$ initially all the door are closed. A person takes $81$ round of the row, numbers as $1^{\text{st }}$ round, $2^{\text{nd }}$ round $\dots \dots ..{81}^{\text{th }}$ round. In each round, he interchage the position of those door number, whose number is multiple of the round number. Find out after ${81}^{\text{st }}$ round, How many doors will be open.

Number of times is the digit $5$ written when listing all numbers from $1$ to ${10}^5$?

Find the number of positive integers less than $2310$ which are relatively prime with $2310.$

The integers from $1$ to $1000$ are written in an order around a circle. Starting at $1,$ every fifteenth number is marked (that is $1, 16, 31,\dots .$ etc.). This process in continued untill a number is reached which has already been marked, then find number of unmarked numbers.

A operation $*$ on a set $A$ is said to be binary, if $x^{*}y\in A,$ for all $x, y\in A,$ and it is said to be commutative if $x^{*}y=y^{*}x$ for all $x, y\in A.$ Now if $A=\left\{a_1, a_2,\dots \dots , a_n\right\},$ then find the following -
(i) Total number of binary operations of $A$
(ii) Total number of binary operation on $A$ such that $a_i^{*}{a}_j\ne a_i^{*}{a}_k$, if $j\ne k$

Find the number of ways of selecting $3$ vertices from a regular polygon of sides $\text{'}2n+1\text{'}$with vertices $A_1, A_2, A_3,\dots \dots , A_{2n+1}$ such that centre of polygon lie inside the triangle.

Let $S=\left\{1,2,3,4\right\}.$ The total number of unordered pairs of disjoint subsets of $S$ is equal to

Number of ways in which $3$ different numbers in $A.P.$ can be selected from $1, 2, 3, \dots \dots .n$ is

The number of three digit numbers of the form $xyz$ such that $x<y$ and $z\le y$ is $N$ then $\left(N-225\right)$ is equal to

A box contains $6$ balls which may be all of different colours or three each of two colours or two each of three different colours. The number of ways of selecting $3$ balls from the box (if ball of same colour are identical) is

in a hockey series between team $X$ and $Y,$ they decide to play till a team wins $\text{'}10\text{'}$ match. Then the number of ways in which a team $X$ wins is $\frac{{}^{20}C_m}{2}$ then $m$ is equal to

The number of ways in which $8$ non-identical apples can be distributed among $3$ boys such that every boy should get atleast $1$ apple & atmost $4$ apples is $N$ then $\left(\frac{N}{60}\right)$ is equal to

Shubham has to make a telephone call to his friend Nisheeth. Unfortunately, he does not remember the $7$ digit phone number. But he remembers that the first three digits are $635$ or $674,$ the number is odd and there is exactly one $9$ in the number. The maximum number of trials that Shubham has to make to be successful is $N$ then $\left(N-3400\right)$ is equal to

Given six line segments of length $2, 3, 4, 5, 6, 7$ units, the number of triangles that can be formed by these segments is

There are$\text{'}n\text{'}$straight line in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the maximum number of fresh lines thus introduced is