Combinations practice · Easy

If ${\;^{189}}{C_{35}} + {\;^{189}}{C_x} = {\;^{190}}{C_x}$, then $x$ is equal to

The total number of positive integral solutions $\left( {x,\,y,\,z} \right),$ such that $xyz = 24$ is

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is

If ${ }^n C_{r-1}=36,{ }^n C_r=84$ and ${ }^n C_{r+1}=126$, then the value of $r$ is equal to

In a college examination, a candidates is required to answer 6 out of 10 questions which are divided into two sections each containing 5 questions. Further the candidate is not permitted to attempt more than 4 questions from either of the section. The number of ways in which he can make up a choice of 6 questions, is

A father with 8 children takes 3 at a time to the zoological garden, as often as he can without taking the same 3 children together more than once. The number of times he will go to the garden, is

Consider 26 tangent lines to an ellipse. The lines separate the plane into several regions, some enclosed and others un- bounded. Then the number of unbounded regions is

There are 20 lines numbered as $1,2,3 \ldots 20$. All the odd numbered lines intersect at a point and all the even numbered lines are parallel. Find the maximum number of point of intersections.

If 7 points out of 12 are in the same straight line, then the number of triangles formed is

Consider the following statements.

I. If there are $n$ different objects, then ${ }^n C_r={ }^n C_{n-r}$, $0 \leq {r} \leq {n}$.

II. If there are $n$ different objects, then ${ }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r, 0 \leq {r} \leq {n}.$

Choose the correct option.

The number of ways in which one can select three distinct integers between 1 and 30, both inclusive, whose sum is even, is

In a Mathematics paper there are three sections containing 4, 5 and 6 questions respectively. From each section 3 questions are to be answered. In how many ways can be selection of questions be made?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?

There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated, is

A lady given a dinner party for six guest. The number of ways in which they may be selected from among ten friends, if two of the friends will not, attends the party together is

The number of ways of selecting 10 balls from unlimited number of red, black, white and green balls is

If ${\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = \frac{k}{{21}},$ then $k$ equals

The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is

The smallest value of $x$ satisfying the inequlity ${\;^{10}}{C_{x - 1}} > 2.{\;^{10}}{C_x}$ is

In the next world cup of cricket there will be 12 teams, divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top teams will qualify for the next round. In this round each team will play against others once.
Four top teams of this round will qualify for the semi- final round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be

The maximum number of points of intersection of 6 circles is

There are 15 points in a plane, no two of which are in a straight line except 4, all of which are in a straight line. The number of triangles that can be formed by using these 15 points is

In a football championship, there were played 153 matches. Every team played one match with each other. The number of teams participating in the championship is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets atleast one ball, is

The number of five digit numbers that contain 7 exactly once is

$A$ is a set containing $n$ elements. A subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is again chosen, the number of ways of choosing so that $(P \cup Q)$ is a proper subset of $A$, is

On the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of a $\triangle \mathrm{ABC}, 3,4,5$ distinct points (excluding vertices A, B, C) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are :

The expression ${\;^n}{C_r} + 4.{\;^n}{C_{r - 1}} + {6.^n}{C_{r - 2}} + 4.{\;^n}{C_{r - 3}} + {\;^n}{C_{r - 4}}$ equals

All possible two factors products are formed from numbers 1, 2, 3, 4,…,200. The number of factors out of the total obtained which are multiples of 5, is

A parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is