Fundamental Principles of Counting

The number of points, at which the two curves $y=\frac{x}{99}$ and $y=\mathrm{sin\pi }x$ intersect, 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

4 buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, then the total possible ways are (bus can't take same path)

There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back, is

Find the number of arrangements of the letters of the word ASSASSINATION.

In how many different ways can the letters of the word "LOGITECH" be arranged in such a way that the vowels always come together?

Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is

Four normal dice are rolled once. The number of possible outcomes in which at least one die shows up $2$ is -

Find the number of arrangements of the letters of the word ASSASSINATION.

Ajay refuses to sit next to either Bharat or Chandan. Deepak refuses to sit next to Enees. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions ?

Three of $128$ objects are labeled $A, B, C.$ You are told that $A$ has rank $1, B$ has rank $2,$ and $C$ has rank $3.$ Let the number of comparisons you need to check the given condition be $N.$ Find the last two digits $N.$

In how many ways can a pack of $52$ cards be divided equally into $4$ groups? What is the number of ways if they are to be divided equally among $4$ players?

Show that the total number of selections that can be made out of the letters of the phrase ‘daddy did a deadly deed’ is $1919$.

Find the total number of selections taking at least one green ball and one blue ball, from $5$ different green balls, $4$ different blue balls and $3$ different red balls.

In a group of $15$ boys there are $7$ boy-scouts. In how many ways can $12$ boys be selected so as to include 

Exactly $6$ boy-scouts, 

Find the number of proper divisors (including $1$) of $3600$.

A candidate has to secure the minimum marks in each of the $5$ subjects to pass an examination. In how many ways can be fail?

Find the number of different rectangles that are there on a chess board.