Division and Distribution into Groups of Equal Sizes

Q. What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these
$\left(i\right)$ four cards are of the same suit.
$\left(ii\right)$ four cards belong to four different suits.
$\left(iii\right)$ are face cards.
$\left(iv\right)$ two are red cards and two are black cards.
$\left(v\right)$ cards are of the same colour?

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. 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. Find the number of permutations of the letters of the word $\text{ALLAHABAD.}$

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. The rank of the word $SUCCESS$, if all possible permutations of the word $SUCCESS$ are arranged in dictionary order is

1. $341$
2. $331$
3. $361$
4. $351$

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?

1. $66$
2. $72$
3. $108$
4. $132$

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. 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. 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 a $5$ digits number is formed using only digits $0,1,3,4,4$, then the correct statements(s) is/are

1. Total possible numbers are $48$
2. Total possible numbers starting from $1$ are $12$
3. the position of the number ${}^{\prime \prime }{41034}^{\prime }$, when arranged in increasing order is $32$
4. the position of the number ${}^{\prime \prime }{34104}^{\prime \prime \prime }$, when arranged in increasing order is $21$

Q. In a football tournament, a team $T$ has to play with each of the $6$ other teams once. Each match can result in a win, draw or loss. Then the number of ways in which the team $T$ finishes with more wins than losses, is

1. $298$
2. $235$
3. $294$
4. $253$

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

Q. Find the principal value of ${\cos }^{-1}(-\frac{1}{2}).$

Q.

The number of ways of dividing 52 cards amongst four players so that three players have 17 cards each and the fourth player just one card, is

1. 

2. 52 !

3. 

4. 

Q. Write the number of ways in which $12$ boys may be divided into three groups of $4$ boys each.

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

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. The number of ways a father can distribute $50$ different coins equally among his $5$ childrens is

1. $\frac{50!\cdot 5!}{(10!{)}^5}$
2. $\frac{50!}{(10!{)}^5\cdot 5!}$
3. $\frac{50!}{(10!{)}^5}$
4. $\frac{50!}{(10!)\cdot 5!}$

Q. Seven different lectures are to be delivered in $7$ periods of a class on a particular day. Out of $7$ lecturers, $A,B$ and $C$ are three of them. If the number of ways in which a routine for the day can be made such that $A$ delivers his lecture before $B$ and $B$ before $C$ is $N,$ then the value of $\left(\frac{N}{120}\right)$ is

Q. There are $80$ families in a small town extension area. $20\%$ of these families own a car each. $50\%$ of the remaining families own a motor cycle each. Let $N$ families in that extension do not own any vehicle. Then the value of $\frac{N}{4}$ is

Q. Let $m$ denote the number of ways in which four different balls of green colour and four different balls of red colour can be distributed equally among $4$ persons if each person has balls of the same colour and $n$ be the corresponding figure when all the four persons have balls of different colour. Then the value of $m+n$ is

Q. Words of length $10$ are formed using the letters $A,B,C,D,E,F,G,H,I,J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x}=$

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. Six boys live in a building of six floors. Ground floor is numbered as $1$ and the top most floor is numbered as $6$. Each floor can accommodate only $1$ person. There are exactly $4$ boys living between A and B. C lives two floors below B. F doesn't live immediately above or below A and B. D is an immediate neighbour of B. How many people live between C and A ?

1. $3$
2. $2$
3. $1$
4. $0$

Q. Let $m$ denote the number of ways in which four different balls of green colour and four different balls of red colour can be distributed equally among $4$ persons if each person has balls of the same colour and $n$ be the corresponding figure when all the four persons have balls of different colour. Then the value of $m+n$ is