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?

From a group of $7$ laptops, $6$ mobile phones and $8$ watches, $4$ gadgets are to be chosen such that there is at least one gadget of each type. In how many ways can it be done?

The letters of the word NATURE are to be arranged so that three vowels should not come together. Find the number of arrangement.

From group of 9 men and 6 women a committee has to be formed of 6 members in which at least 3 women are required. Find the number of ways to form the committee.

There are $17$ points in a plane out of which $6$ are collinear. Find the number of lines that can be formed from $17$ points.

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

Find the number of arrangements of the letters of the word ASSASSINATION.

In how many different ways can the letters of the word "LOGITECH" be arranged in such a way that the vowels always come together?

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