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 numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits $0,1,2,5,9$, if the repetition of the digits is allowed, is $\_\_\_\_$

If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which $C$ and $S$ do not come together, is $(6!)k$then$k$ is equal to

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______?

Let S denote the set of 4-digit numbers $a b c d$ such that $a>b>c>d$ and P denote the set of 5 -digit numbers having product of its digits equal to 20. Then $n(\mathrm{~S})+n(\mathrm{P})$ is equal to $\_\_\_\_$

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

Eight persons are to be transported from city$A$ to city$B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1,2,3,4,5$ with repetition, is ________

Let $f(x)=2x^{n}+\lambda\,,\lambda\,\in\,\mathbb{R},n\in\,\mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

Number of ways of arranging $8$identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________