Division and Distribution into Groups of Equal Sizes

Q. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5, and 7. The smallest value of n for which this is possible is

1. 6
2. 7
3. 8
4. 9

Q. Five balls are to be placed in three boxes. Each box can hold all the five balls so that no box remains empty.

If balls as well as boxes are identical but boxes are kept in a row then number of ways is

Q. The number of ways $10$ boys can be divided into $2$ groups of $5$, such that two particular boys are in the different groups, is

1. $70$
2. $35$
3. $252$
4. $126$

Q. Five balls are to be placed in three boxes. Each box can hold all the five balls so that no box remains empty.

If balls are different but boxes are identical then number of ways is

Q. Number of ways of dividing $80$ cards into $5$ equal groups of $16$ each is :

1. $\frac{80!}{(16!{)}^5}$
2. $\frac{80!}{(5!{)}^{16}}$
3. $\frac{80!}{(5!{)}^5}$
4. $\frac{80!}{(16!{)}^5\cdot 5!}$

Q. The number of ways a pack of $52$ cards can be divided among four players in $4$ sets, three of them having $17$ cards each and the fourth one just $1$ card is :

1. $\frac{52!}{(17!{)}^3}\times 4$
2. $\frac{52!}{3\cdot (17!{)}^3}$
3. $\frac{52!}{3!(17!{)}^3}$
4. $\frac{52!}{(3!{)}^3(17!)}$

Q. A rectangle with side lengths as $2m-1$ and $2n-1$ units is divided into squares of unit length by drawing parallel lines as shown in diagram, then the number of rectangles possible with odd side length is

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

Q. If the set $S=\{1,2,3,\cdots ,12\}$ is to be partitioned into three sets $A,B,C$ of equal size such that $A\cup B\cup C=S,A\cap B=B\cap C=A\cap C=\varphi$ then the number of ways of partitioning $S$ is :

1. $\frac{12!}{3!(4!{)}^3}$
2. $\frac{12!}{3!(3!{)}^4}$
3. $\frac{12!}{(3!{)}^4}$
4. $\frac{12!}{(4!{)}^3}$

Q. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5, and 7. The smallest value of n for which this is possible is

1. 6
2. 7
3. 8
4. 9

Q. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5, and 7. The smallest value of n for which this is possible is

1. 8
2. 6
3. 7
4. 9