Division and Distributuion into Groups of Unequal Sizes.

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

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. ${}^{10}{C}_4$
2. ${}^{11}{C}_4$
3. ${}^{12}{C}_4$
4. ${}^9{C}_4$

Q.

The number of ways in which 5 identical balls can be kept in 10 identical boxes, if not more than one can go into a box, is

1. ${}^{10}{P}_5$

2. $(\frac{10}{5})$

3. 5

4. 1

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 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

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. A committee of 5 is to be formed from 6 boys and 5 girls, . The number of ways that the committee can be formed so that the committee contains at least one boy and one girl is

1. 440
2. 445
3. 450
4. 455

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