Fundamental Principle of Counting

How many permutations can be made out of the letters in the word island taking four letters at a time?

Five friends Lalit, Feroz, Shahid, John and Manjeet are painter, Singer, Dancer, Poet or Sculptor. Lalit and Feroz are not Sculptor or Dancer. John and Manjeet are not Poet or Painter. Shahid is neither Painter nor Dancer. Manjeet is not a Dancer. Feroz and Shahid are not Poet or Singer. Who is Poet?

Five different Mathematics books,$4$ different electronics books, and $2$different communications books are to be placed on a shelf with the books of the same subject together. Find the number of ways in which the books can be placed?

A motorist knows four different routes from Bristol to Birmingham. From Birmingham to Sheffield he knows three different routes and from Sheffield to Carlisle he knows two different routes. How many routes does he know from Bristol to Carlisle ?

In how many ways can $4$ boys and $4$ girls be seated alternately in a row of $8$ seats?

Show that $\frac{12!}{5!7!}+\frac{12!}{6!6!}=\frac{13!}{6!7!}$

The number of $3$-digit odd numbers divisible by $3$ that can be formed using the digits $1,2,3,4,5,6$ when repetition is not allowed is

The exponent of $6$ in $72!$ is

There are $15$ students in a group and among them $9$ are boys and the remaining are girls. From this group $5$ students to be selected for a competition and at least $3$ should be girls. In how many ways can it be done?

Let $E\left(n\right)$ denotes the sum of the even digits of $n$. For example $E\left(1243\right)=2+4=6$, then the value of $E\left(1\right)+E\left(2\right)+E\left(3\right)+\mathrm{.....}+E\left(100\right)$ is equal to

Among the inequalities below, which ones are true for all natural numbers $n$ greater than $1000$ ?
I. $n!\le n^n$
II. ${\left(n!\right)}^2\le n^n$
III. ${10}^n\le n!$
IV. $n^n\le \left(2n\right)!$

The largest power of $2$ that divides $\frac{200!}{100!}$ is

Consider $5$ straight lines in a plane such that no two of them are parallel and no three of the them intersect at a point. Then, the number of disjoint regions into which the plane is divided by these lines equals to

Which of the following statements is correct regarding the statement given below?

The five-digit numbers can be formed from the first four prime numbers.

For a given matrix, let $R_i$, denote the sum of all entries in its $i^{th}$ row and $C_j$ denote the sum of all entries in its $j^{th}$ column. How many $3\times 3$ matrices with non-negative integer entries are there such that $R_1=R_2=C_1=C_2=2$ and $R_3=C_3=1?$

Three varieties of mangoes and four varieties of apple are available for shakes in a juice centre. More than one mango can be used in one shake but same is not the case with apple. Only one apple can be used in one shake. However, apples and mangoes can also be mixed to make shakes. Find the total number of shaes available in the juice center.

The number of different solutions of the equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=12$, where each of $\mathrm{x},\mathrm{y}$ and $\mathrm{z}$ is a positive integer, is

There are six teachers. Out of them, two teach physics, other two teach chemistry and the rest two teach Mathematics. They have to stand in a row such that Physics,Chemistry and Mathematics teachers are always in a set. The number of ways in which they can do, is: