Division and Distributuion into Groups of Unequal Sizes.

Q. In a high school, a committee has to be formed from a group of $6$ boys $M_1,M_2,M_3,M_4,M_5,M_6,$ and $5$ girls $G_1,G_2,G_3,G_4,G_5.$
(i) Let ${\alpha }_1$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, having exactly $3$ boys and $2$ girls.
(ii) Let ${\alpha }_2$ be the total number of ways in which the committee can be formed such that the committee has atleast $2$ members, and having an equal number of boys and girls.
(iii) Let ${\alpha }_3$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, atleast $2$ of them being girls.
(iv) Let ${\alpha }_4$ be the total number of ways in which the committee can be formed such that the committee has $4$ members, atleast $2$ girls and such that both $M_1$ and $G_1$ are NOT in the committee together.
$\begin{array}{cccc}& LIST-I& & LIST-II\\ P.& \text{The value of}{\alpha }_1\text{is}& 1.& 136\\ Q.& \text{The value of}{\alpha }_2\text{is}& 2.& 189\\ R.& \text{The value of}{\alpha }_3\text{is}& 3.& 192\\ P.& \text{The value of}{\alpha }_4\text{is}& 4.& 200\\ & & 5.& 381\\ & & 6.& 461\end{array}$
The correct option is:

1. $P\to 4;Q\to 6;R\to 2;S\to 1$
2. $P\to 1;Q\to 4;R\to 2;S\to 3$
3. $P\to 4;Q\to 2;R\to 3;S\to 1$
4. $P\to 4;Q\to 6;R\to 5;S\to 2$

Q. Number of permutations of $1,2,3,4,5,6,7,8$ and $9$ taken all at a time such that
$1$ appears to the left of $2$
$3$ appears to the left of $4$ and
$5$ appears to the left of $6$ is $k\times 7!$, then the value of $k$ is

Q.

Number of pairs of positive integers $(p,q)$ whose LCM (Least common multiple) is $8100$, is “$K$”. Then number of ways of expressing $K$ as a product of two co-prime numbers is ...............

1. 4
2. 8
3. 2
4. 6

Q. Let $A=\{1,2,3,4,5,6,7\}.$ The number of surjective functions defined from $A$ to $A$ such that $f\left(i\right)=i$ for atleast four values of $i$ from $i=1,2,\cdots ,7,$ is

Q. There are three piles of identical red, blue and green balls and each pile contains at least $10$ balls. The number of ways of selecting $10$ balls if twice as many red balls as green balls are to be selected, is

1. $4$
2. $8$
3. $6$
4. $3$

Q. Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number. The numbers of such numbers if
$x_1<x_2<x_3\le x_4<x_5<x_6$ is

1. ${}^9{C}_3$
2. ${}^{10}{C}_4$
3. ${}^{10}{C}_3$
4. ${}^9{C}_4$

Q. The number of ways $5$ objects be divided among the persons $A$ and $B$ if

1. no one gets all the objects is $30$
2. one can get all the object is $30$
3. one can get all the object is $32$
4. no one gets all the objects is $31$

Q. The number of ways in which $5$ different toys can be distributed to $3$ children if each child can get any number of toys, is also equal to

1. Number of subsets of $A=\{1,2,3,4,5,6,7,8,9,10\}$ which contain $2$ elements of $A$ such that their sum not equal to $11$
2. Number of non-negative integral solutions of the equation $xyz=2310$
3. Number of $6-$digit numbers less than $200000$ formed by using only the digits $1,2$ and $3$
4. Number of all possible selections of one or more questions from $5$ given questions, with an alternative question corresponding to each question

Q. The total number of $4-$letter words that can be made by using the letters of word $TOMATO$ is

Q. A committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?

Q. The number of zero(s) in ${}^{500}{C}_{20}$ is

Q.

In how many ways five different balls be arranged so that two particular balls are never together ?

Q. If $C_0,C_1,C_2\cdots$ are combinational coefficient in the expansion of $(1+x{)}^n;n\in N$ and $(C_0+C_1)\cdot (C_1+C_2)\cdot (C_2+C_3)\cdots (C_{19}+C_{20})=\frac{C_0\cdot C_1\cdot C_2\cdots C_{18}\cdot a^{20}}{b!};(a,b\in N)$, then the value of $(a+b)$ is

Q. The number of ways $5$ objects be divided among the persons $A$ and $B$ if

1. no one gets all the objects is $30$
2. one can get all the object is $30$
3. one can get all the object is $32$
4. no one gets all the objects is $31$

Q. There are three piles of identical red, blue and green balls and each pile contains at least $10$ balls. The number of ways of selecting $10$ balls if twice as many red balls as green balls are to be selected, is

1. $3$
2. $4$
3. $6$
4. $8$

Q. From a committee of $8$ persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?

Q. If a class consists of $6$ periods, then the number of ways in which $5$ subjects can be taught such that each subject must be allotted at least one period and no period remains vacant is

1. $\frac{6!}{2!}$
2. $5!\times \frac{6!}{2!}$
3. $5\times \frac{6!}{2!}$
4. $5\times 6!$

Q. Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number.The number of such numbers if
$x_1<x_2<x_3<x_4<x_5<x_6$ is

1. ${}^9{C}_3$
2. ${}^{10}{C}_3$
3. ${}^9{C}_5$
4. ${}^{10}{C}_6$

Q. Find the number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty.

Q. A man has $3$ friends. If $N$ is the number of ways he can invite one friend everyday for dinner on $6$ successive nights so that no friend is invited more than $3$ times, then the value of $N/170$ is

Q. Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards, then the mean of the number of queens is

1. $\frac{2}{23}$
2. $\frac{2}{13}$
3. $\frac{1}{13}$
4. $\frac{3}{13}$

Q. There are $8$ events that can be scheduled in a week. Then total number of ways that these $8$ events are scheduled on exactly $6$ days of a week is given by $266\times k!,$ where $k\in N.$ The value of $k$ is

1. $7$
2. $8$
3. $5$
4. $6$

Q. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball, is ?___

1. 75
2. 150
3. 210
4. 243

Q. The total number of ways of dividing $15$ different things into groups of $8,4$ and $3$ respectively is

1. ${}^{15}{C}_8\cdot {}^7{C}_4$
2. ${}^{15}{C}_8\cdot {}^8{C}_4$
3. ${}^{15}{C}_6\cdot {}^7{C}_4$
4. ${}^{15}{C}_6\cdot {}^8{C}_4$

Q.

The number of ways in which 10 persons can go in two boats so that there may be 5 on each boat, supposing that two particular persons will not go in the same boat is_______.

1. 

2. 

3. 

4. 

Q. Assertion :$e^{\pi }>{\pi }^e$ Reason: The function $f\left(x\right)=x^{\frac{1}{x}}$ attains global maxima at x = e.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. No. Of ways in which 12 identical balls can be put in 5 different boxes in a row, if no box remains empty is

1. 
2. 
3. 
4. 

Q. An urn contains $5$ red and $2$ black balls. Two balls are randomly selected. If $X$ represents the number of black balls, what are the possible values of $X$?

Q. If we divide a two-digit number by the sum of its digits, we get $4$ as a quotient and $3$ as a remainder. Now if we divide that two-digit number by the product of its digits, we get $3$ as a quotient and $$5 as a remainder and the two-digit number.

Q. The position vector of a point P is $\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$ where x and y are positive integers and $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ If $\overrightarrow{r}.\overrightarrow{a}=10$, then the number of possible positions of P is :

1. 48
2. 72
3. 24
4. 36