Combination

An examination paper consists of$12$ questions divided into parts$A$ &$B$. Part-$A$ contains$7$ questions & Part-$B$ contains$5$ questions. A candidate is required to attempt$8$ questions selecting$3$ at least from each part. In how many maximum ways can the candidate select the questions?

The total number of paths from the origin to the point $\left(9,9\right)$ in the cartesian plane which never pass through $\left(5,5\right)$, where paths consisting only of steps going $1$ unit North or $1$ unit East, is

Let $A$ be the set of all permutations $a_1,a_2,\dots ,a_6$ of $1,2,\dots ,6$ such that $a_1,a_2,\dots ,a_k$ is not a permutation of $1,2,\dots ,k$ for any $k, 1\le k\le 5$. Then, the number of elements in $A$ is

If total number of students is $10$ and number of students to be selected or permutated is $5$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $6$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $15$ and number of students to be selected or permutated is $7$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $12$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

There are $4, 5$ and $6$ points on the three parallel and co-planar lines. If no set of four or more points is cyclic and no set of three points lying on distinct lines is collinear, then the number of circles passing through exactly three points are

An urn contains $5$ red, $4$ black and $3$ white marbles. Then, the number of ways in which $4$ marbles can be drawn from it so, that at most $3$ of them are red, is

Statement 1: If a polygon has total $66$ diagonals, then the total number of its sides are $12$.

Statement 2: The total number of ways of selecting $2$ points out of $n$ non-collinear points is ${}^n\mathrm{C}_2$.

In how many different ways can the letters of the word DESIGN be arranged so that the vowels are at the two ends?

A password is set with $3$ distinct letters from the word LOGARITHMS. How many such passwords can be formed?

The number of diagonals of a polygon with $15$ sides is

There are $5$ bananas and $4$ oranges. In how many different ways fruits can be selected if fruits of same kind are different? (if minimum $1$ fruit is selected)

In a cricket championship there are $36$ matches. The number of teams, if each plays $1$ match with other are

In how many ways a team of $5$ members can be chosen from 8 members:

The number of diagonals of $15$ sided polygon are _____.