Permutation Combination

Consider a rectangle $ABCD$having $5,6,7,9$ points in the interior of the line segments $AB,BC,CD,DA$ respectively. Let $\alpha\,$ be the number of triangles having these points from different sides as vertices and…

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

The number of seven digits odd numbers, that can be formed using all the seven digits $1,2,2,2,3,3,5$ is

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is $\qquad$ -

The number of ways of getting a sum 16 on throwing a dice four times is______

Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text {th }}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at…

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$apples, is

Let $S=\left{1,2,3,5,7,10,11\right}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is _____ .

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{\mathrm{n}}$; $m, n \in\{1,2,3, \ldots, 9\}$, then $m+n$ is equal to _______

In an examination of Mathematics paper, there are $20$questions of equal marks and the question paper is divided into three sections : $A,B$ and $C$. A student is required to attempt total $15$ questions taking at least…

In an examination, $5$ students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in \mathrm{S}_6$, but $\{1,2,4\} \notin \mathrm{S}_6$. Then…