Combinations

Study the given information carefully and answer the questions that follow—
A committee of $6$ teachers is to be formed out of $5$ History teachers, $6$ English teachers and $4$ Economics teachers. In how many ways can the committee be formed if -

How many teacher can be included in the committee?
If $6$ teachers have to selected out of $15$

Study the given information carefully and answer the question that follow—
A committee of $6$ teachers is to be formed out of $5$ History teachers, $6$ English teachers and $4$ Economics teachers. In how many ways can the committee be formed if -
$2$ teachers from each stream are to be included.
(1)number of teachers in class $A$ is $6$
(2) Number of teacher in class $B$ is $5$
(3)Number of teacher in class $C$ is $4$

Study the given information carefully and answer the questions that follow—
A committee of $6$ teachers is to be formed out of $5$ History teachers, $6$ English teachers and $4$ Economics teachers. In how many ways can the committee be formed if -

At least $1$ teacher from History stream is to be included?

Study the given information carefully and answer the questions that follow—
A committee of $6$ teachers is to be formed out of $5$ History teachers, $6$ English teachers and $4$ Economics teachers. In how many ways can the committee be formed if -

If $6$ teachers has to be selected out of $11$ teachers the number of ways to select?

If ${}^{n}P_r=66$ and ${}^{n}C_r=11$ find the vaue of 'r'

There are $12$ players in a stadium for a match in how many ways the first two prizes may be won?

There are $5$ letter E,F,G,H,I and it must be arrange in order that G always comes before H and after F. In how many ways it can be done?

There are two types of rooms available in a hotel-one single and one double. In how many ways $5$ persons can accomodate in these rooms?

How many number can be formed from the digit $1,2,3$ with no rejection ?

In an entrance examination a candidate is required to answer $5$ multiple choice question in which he answer all in correctly. In how many ways it can be done?

The floor is to be pared with triangular tiles. In how many ways i can be done if the vertices of the bottom are at a distance of $10$ points and $5$ points lies within the lines.

A group of $5$ members must be selected from $3$ man and $5$ women such that the group must not involve more than $2$ men. In how many ways its can be done?

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 ${}^n\mathrm{C}_2$.

The value of the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2 {}^{n}C_r$ is, where ${}^{n}C_r$ is the number of selection of $r$ things out of $n$ things

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

In a group of $6$ boys and $4$ girls, a team consisting of four children is formed such that the team has atleast one boy. The number of ways of forming a team like this is

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

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)

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