Combinations practice · Mixed difficulty

From 12 books, the difference between number of ways a selection of 5 books when one specified book is always excluded and one specified book is always included, is

If ${\;^{35}}{C_{n + 7}} = {\;^{35}}{C_{4\;n - 2}}$, then all the values of $n$ are given by

The number of ways in which two teams $A$ and $B$ of 11 players each can be made up from 22 players so that two particular players are on the opposite sides is

The interior angles of a regular polygon measure ${160^0}$ each. The number of diagonals of the polygon are

The number of integral points on the hyperbola $x^{2}-y^{2}=(2000)^{2}$ is (an integral point is a point both of whose co-ordinates are integer)

The straight lines ${I_1},{I_2},{I_3}$ are parallel and lie in the same plane. A total numbers of $m$ points are taken on ${I_1},n$ points on ${I_2},k$ points on ${I_3}$. The maximum number of triangles formed with vertices at these points is

The number of times the digit 3 will be written when listing the integers from 1 to 1000 , is

If $\sum\limits_{{r=1}}^n\left[\log _2 r\right]=1898$ (where [.] denotes greatest integer function), then number of zeroes at the end of $\mathrm{n}$ is/are

($n$!)! is divisible by

The number of ways in which 12 balls can be divided between two friends, one receiving 8 an the other 4, is

A class consists of 4 boys and $g$ girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being sent every week. During, the picnic, the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of $g$ is

Out of 6 boys and 4 girls, a group of 7 is to be formed. In how many ways can this be done, if the group is to have a majority of boys?

The number of group that can be made from5 different green balls, 4 different blue balls and 3 different red balls, if at latest 1 green and 1 blue ball is to be included, is

Sita has 5 coins each of the different denomination. The number different sums of money she can form is

Statement 1: If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q>p$. Then the total number of functions from set $A$ to set $B$ is $q^{p}$.
[Online May 12, 2012]
Statement 2: The total number of selections of $p$ different objects out of $q$ objects is ${ }^{q} \mathrm{C}_{p}$.

A person wishes to make up as many different parties as he can out of his 20 friends such that each party consists of the same number of persons. The number of friends he should invite at a time, is

The value of ${ }^{50} C_4+\sum\limits_{{r=1}}^6{ }^{56-r} C_3$ is

${\;^n}{C_r} + 2{\;^n}{C_{r - 1}} + {\;^n}{C_{r - 2}}$ is equal to

Number of diagonals of a polygon are 65 , then the number of sides of polygon are

The value of ${ }^{50} C_{4}+\sum_{r=1}^{6}{ }^{56-r} C_{3}$ is

Given 5 different green dyes, 4 different blue dyes and 3 different red dyes. The number of combinations of dyes that can be chosen by taking at least one green and one blue dye is

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n+1}-T_n=21$, then $n$ equals

In a plane there are 37 straight lines, of which 13 pass through the point $A$ and 11 pass through the point $B$. Besides, no three lines pass through one point, no line passes through both points $A$ and $B$ and no two are parallel, the number of intersection points the lines have, is

The straight lines $I_1, I_2, I_3$ are parallel and lie in the same plane. A total number of $m$ points on $I_1 ; n$ points on $I_2 ; k$ points on $I_3$, the maximum number of triangles formed with vertices at these points is

There are $n$ points in a plane, no three of which are collinear except ' $p$ ' points all of which are on a line. How many
(a) straight lines can be formed and
(b) triangles can be formed out of these $n$ points?

Find the number of three digit numbers which can be formed using the digits $0,1,2,2,3, 3,3 .$

Out of 10 consonants and 4 vowels, words with 6 consonants and 3 vowels are formed. The number of such words is

The number of ways of selecting 4 cards of an ordinary pack of playing cards so that exactly 3 of them are of the same denomination is

The number of diagonals of a polygon of 15 sides is