Combination

Q. If a polygon has 44 diagonals then the number of its sides are?

1. 9
2. 10
3. 11
4. 12

Q.

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is:

1. $262$
2. $190$
3. $225$
4. $157$

Q.

If $9999=4$, $8888=8$, $1816=6$, $1212=0$, then, $1919=$?

Q.

The sides $\mathrm{AB},\mathrm{BC},\mathrm{CA}$ of a triangle $\mathrm{ABC}$ have respectively $3,4$ and $5$ points lying on them.

The number of triangles that can be constructed using these points as vertices is?

1. $205$

2. $220$

3. $210$

4. None of these

Q. Consider a convex polygon which has $35$ diagonals. Then match the following columns:
​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{a. Number of triangles joining the vertices of the}& \\ \text{polygon}& \text{p.}210\\ \text{b. Number of points of intersections of diagonal}& \\ \text{which lies inside the polygon}& \text{q.}120\\ \text{c. Number of triangle in which exactly one side}& \\ \text{is common with that of polygon}& \text{r.}10\\ \text{d. Number of triangles in which exactly two sides}& \\ \text{are common with that of polygon}& \text{s.}60\end{array}$

1. $a\to q;b\to p;c\to s;d\to r.$
2. $a\to s;b\to r;c\to q;d\to p.$
3. $a\to q;b\to s;c\to p;d\to r.$
4. $a\to s;b\to q;c\to r;d\to p.$

Q. There are three bags $B_1,B_2,\text{and}{B}_3$ containing $2$ red & $3$ white, $5$ red & $5$ white, $3$ red & $2$ white balls respectively. A ball is drawn from bag $B_1$ and placed in $B_2$. Then a ball is drawn from bag $B_2$ and placed in $B_3$. Then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed, if same colour balls are used in the first and the second transfers is (assuming all balls are different)

1. $30$
2. $180$
3. $150$
4. $25$

Q. In how many ways a team of 11 players can be selected from 15 players if-

• captain is always included in the team?
• captain is not included in the team?

1. $14C11,14C11$
2. $15C10,15C11$
3. $14C10,15C11$
4. $14C10,14C11$

Q. If m parallel lines in plane are intersected by n parallel lines, then number of parallelograms formed is

1. $\frac{m!n!}{{(2!)}^2}$
2. $\frac{m!n!}{(m-2)!(n-2)!}$
3. $\frac{m!n!}{{(2!)}^2(m-2)!(n-2)!}$
4. $\frac{(m+n)!}{(m+n-2)!2!}$

Q. The number of ways of arranging 6 boys and 6 girls in a row so that boys and girls come alternatively

1. 6! 6!
2. 7! 6!
3. 7.6! 6!
4. 2.6! 6!

Q.

Consider the following two statements :

Statement-$1$ : The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is ${}^{9}C_3$.

Statement-$2$ : The number of ways of choosing any $3$ places from $9$ different places is ${}^{9}C_3$.

Then which of one of the following choices is correct?

1. Both Statement-$1$ and Statement-$2$are true and Statement-$2$ is a correct explanation for Statement-$1$.

2. Both Statement-$1$ and Statement-$2$are true and Statement-$2$ is not a correct explanation for Statement-$1$

3. Statement-$1$ is true but Statement-$2$ is false.

4. Both Statement-$1$ and Statement-$2$ are true.

Q. 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

1. 6
2. 11
3. 13
4. 12

Q. A bag contains 9 balls marked with the digits 1, 2, 3.....9. If two balls are drawn from the bag then find the number of ways of getting the sum of digits on the balls as odd?

1. $4C_1\times 5C1$
2. $5C1\times 4C2$
3. $4C1\times 4C1$
4. $4C1\times 5C2$

Q. Two teams are playing a series of five matches between them. Any random match ends with three results i.e, win, loss or draw for a team. Let a group of $n$ people forecast the result of a particular team for each match and no two people make the same forecast for the series of matches. The maximum number of people required in a group such that a person forcasts all the results correctly for all the matches is

Q. The number of 4-digit numbers that can be formed by using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 such that the At least digit used is 4, when repetition of digits is allowed, is

1. 617
2. 671
3. 716
4. 761

Q. $f\left(n\right)=\sum _{r=1}^n[r^2\left({}^n{C}_r{-}^n{C}_{r-1}\right)+(2r+1\left){(}^n{C}_r\right)],\text{then}$

1. $f\left(30\right)=960$
2. $f\left(21\right)=483$
3. $f\left(16\right)=64$
4. $f\left(11\right)=44$

Q. In an examination, question paper consists of $15$ questions with two sections $A$ and $B$ containing $6$ and $9$ questions respectively. In how many ways can a student attempt $8$ questions, selecting at least $3$ questions from each section ?

1. $5126$
2. $4914$
3. $3600$
4. $4832$

Q. If the sides $AB$, $BC$ and $CA$ of a $△ABC$ have $a$, $b$ and $c$ points lying on them respectively, then the number of triangles that can be constructed using these points as vertices, if

1. two vertices lie on same side $={}^{a+b+c}{C}_3-({}^a{C}_3+{}^b{C}_3+{}^c{C}_3+abc)$
2. two vertices lie on the same side $={}^{a+b+c}{C}_3-abc$
3. all the vertices lie on different sides $=abc$
4. all the vertices lie on different sides $={}^{a+b+c}{C}_3-({}^a{C}_3+{}^b{C}_3+{}^c{C}_3)$

Q. There are $12$ persons in a party, and if each two of them shake hands with each other, then how many hand shakes happen in the party?

Q. The number of ways of distriuting 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

1. 5
2. ${}^8{C}_3$
3. $3^8$
4. 21

Q. Total number of polynomials of the form $x^3+a{x}^2+bx+c,$ that are divisible by $x^2+1,$ where $a,b,c\in \{1,2,3,\dots ,9,10\}$ is

1. $8$
2. $10$
3. $15$
4. $30$

Q. Number of combination of different things taking r things at a time when,

1. $n-P-QCr-P$
2. $n-PCr-P$
3. $n-PCr$

Q.

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

1. 269

2. 300

3. 271

4. 302

Q. If the sides $AB$, $BC$ and $CA$ of a $△ABC$ have $a$, $b$ and $c$ points lying on them respectively, then the number of triangles that can be constructed using these points as vertices, if

1. two vertices lie on same side $={}^{a+b+c}{C}_3-({}^a{C}_3+{}^b{C}_3+{}^c{C}_3+abc)$
2. two vertices lie on the same side $={}^{a+b+c}{C}_3-abc$
3. all the vertices lie on different sides $=abc$
4. all the vertices lie on different sides $={}^{a+b+c}{C}_3-({}^a{C}_3+{}^b{C}_3+{}^c{C}_3)$

Q. There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$, then the value of $m$ is :

1. $7$
2. $9$
3. $11$
4. $12$

Q. The number of different ways in which a committee of $4$ members formed out of $6$ Asians, $3$ Europeans and $4$ Americans if the committee is to have at least one member from each of the regional groups, is

1. $320$
2. $340$
3. $360$
4. $380$

Q. Statement-1; The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is $9C_3$.
Statement-2: The number of ways of choosing any 3 places from 9 different places is $9C_3$.

1. Statement-1 is true, Statement -2 is true: Statement-2 is not a correct explanation for Statement-1.
2. Statement-1 is true, Statement-2is false.
3. Statement-1 is false, Statement-2 is true.
4. Statement-1 is true, Statement-2 true; Statement-2 is a correct explanation for Statement-1.

Q. Total 4 digit odd numbers that can be formed, if the digits used is not to be repeated again is

1. 2240
2. 2420
3. 2440
4. 2520

Q.

There are 16 points in a plane out of which 6 are collinear, then how many lines can be drawn by joining these points

1. 106

2. 105

3. 60

4. 55

Q. Two teams are playing a series of five matches between them. Any random match ends with three results i.e, win, loss or draw for a team. Let a group of $n$ people forecast the result of a particular team for each match and no two people make the same forecast for the series of matches. The maximum number of people required in a group such that a person forcasts all the results correctly for all the matches is