Dearrangement

Q. Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements,
$\left(i\right)$ do the words start with $P$
$\left(ii\right)$ do all the vowels always occur together
$\left(iii\right)$ do the vowels never occur together
$\left(iv\right)$ do the words begin with $I$ and end in $P$ ?

Q.

Number of $3\times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ is

1. Less than $4$

2. $5$

3. $6$

4. at least $7$

Q. There are five students $S_1,S_2,S_3,S_4$ and $S_5$ in a music class and for them, there are five seats $R_1,R_2,R_3,R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i,$ $i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.

The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is

1. $\frac{3}{40}$
2. $\frac{1}{8}$
3. $\frac{7}{40}$
4. $\frac{1}{5}$

Q. There are five students $S_1,S_2,S_3,S_4$ and $S_5$ in a music class and for them there are five seats $R_1,R_2,R_3,R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i,i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.

The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is

1. $\frac{3}{40}$
2. $\frac{1}{8}$
3. $\frac{7}{40}$
4. $\frac{1}{5}$

Q. There are six balls of different colours and six boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed in the boxes such that atmost two balls are in their corresponding colour boxes is equal to

Q. The number of words (with or without meaning) that can be formed from all the letters of the word ${}^{\prime \prime }LETTE{R}^{\prime \prime }$ in which vowels never come together is

Q. A man has $7$ letters for his $7$ friends. The letter are kept in the envelopes at random. The number of ways in which exactly $3$ letters are going to correct envelope and rest $4$ letters are going to the wrong envelopes is

1. $315$
2. $350$
3. $420$
4. $210$

Q. There are $5$ people and $5$ different pairs of shoes. If each person chooses at random one right shoe and one left shoe how many combinations can be made in which no one gets a correct pair.

1. $44$
2. $120$
3. $5280$
4. $2880$

Q. A person writes letters to $6$ friends and addresses the corresponding envelopes. The number of ways in which $5$ letters can be placed in wrong envelopes is

Q. Number of $4$ digit numbers of the form
$N=abcd,$ which satisfy following conditions :
$\left(i\right)4000\le N<6000$
$\left(ii\right)N$ is a multiple of $5$
$\left(iii\right)3\le b<c\le 6$ is equal to

1. $16$
2. $4$
3. $12$
4. $24$

Q. The number of ways in which $5$ balls numbered $1-5$ can be placed in $5$ boxes numbered as $1-5$ (each box can accomodate one ball only) when

1. all the balls should be kept on correspnding numbered boxes is $5$
2. one ball is not kept in correspnding numbered box is $1$
3. three balls are not kept in correspnding numbered boxes is $20$
4. four balls are not kept in correspnding numbered boxes is $45$

Q. The number of ways in which all the letters of the word $HOUSE$ can be arranged such that no letter will occur in its original position is $11k$, then the value of $k$ is

Q. There are four balls of different colors and four boxes of colors same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own color is ___.

Q. The value of sum of the given determinants,
$|A|=\left|\begin{array}{ccc}103& 115& 114\\ 111& 108& 106\\ 104& 113& 116\end{array}\right|,|B|=\left|\begin{array}{ccc}113& 116& 104\\ 108& 106& 111\\ 115& 114& 103\end{array}\right|$ is

Q. The total number number of ways in which all the digits of the number $79853$ can be arranged such that no digit will occur in its original position as given in the number is

Q.

The number of ways in which all the letters of the word $GARDEN$ can be arranged such that no letter is in its original position is

Q. The letters of the word "$INDEPENDENCE$" are arranged in all possible ways. Of these the number of words in which the '$D$'s come together is

1. $11!$
2. $\frac{11!}{4!3!}$
3. $\frac{11!}{4!}$
4. $\frac{11!}{4!3!2!}$

Q. $\sum _{r=1}^n\frac{r}{1,3,5,\mathrm{7..........}(2r+1)}$ is equal to

1. $\frac{1}{2}[1-\frac{1}{1,3,\mathrm{5..............}(2n-1)}]$
2. $\frac{1}{4}[1-\frac{1}{1,3,\mathrm{5..............}(2n-1)}]$
3. $\frac{1}{4}[1+\frac{1}{1,3,\mathrm{5..............}(2n-1)}]$
4. none of these

Q.

5 countries president and prime minister went for summit. If all 5 prime minister shake hand to president at random, in how many ways they can shake hand if none of the same country PM shake hand to that country president.

1. 120

2. 24

3. 44

4. 9

Q. A can solve 75% of the problems of Mathematics and B can solve 70%. What is the probability that either A or B can solve a problem chosen at random ?

1. $\frac{37}{40}$
2. $\frac{21}{40}$
3. $\frac{19}{40}$
4. $\frac{3}{40}$

Q.

Find the total number of selections of A + B + C + D objects where A are alike (of one kind), B are alike (of second kind), C are alike (of third kind) and D are alike (of fourth kind) if he has to select at least one of each kind of object.

1. A + B + C + D

2. (A +1) (B + 1)(C +1) (D+1)

3. 

4. (A +1) (B + 1)(C +1) (D+1) - 1

Q. A person writes letters to $6$ friends and addresses the corresponding envelopes. The number of ways in which $5$ letters can be placed in wrong envelopes is

Q. In an examination, the probability of a candidate solving a question is $\frac{1}{2}$. Out of given $5$ questions in the examination, what is the probability that the candidate was able to solve at least $2$ questions?

1. $\frac{3}{16}$
2. $\frac{1}{64}$
3. $\frac{1}{2}$
4. $\frac{13}{16}$

Q. There are six balls of different colours and six boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed in the boxes such that atmost two balls are in their corresponding colour boxes is equal to

Q. Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

1. 264
2. 265
3. 53
4. 67

Q.

5 countries president and prime minister went for summit. If all 5 prime minister shake hand to president at random, in how many ways they can shake hand if none of the same country PM shake hand to that country president.

1. 9

2. 44

3. 120

4. 24

Q. All the letters of the word $EAMCET$ are arranged in all possible ways. The number of such arrangements in which no two vowels are adjacent to each other is

1. $36$
2. $54$
3. $72$
4. $144$

Q. A person writes letters to six friends and addresses the corresponding envelopes. Let $x$ be the number of ways so that at least two of the letters are in wrong envelopes and $y$ be the number of ways so that all the letters are in wrong envelopes, then $x-y=$

1. 716
2. 454
3. 265
4. 0

Q. There are four parcels and five post-offices. In how many different ways can the parcels be sent by post?

Q. There are $24$ team participating in a mathematics competition from a different school, one team per class. The competition is from $VIII-X$ classes. Find the probability of:
Coming first from each class per school ?
Getting a prize, if there are three prizes per class $I,II$ and $III$ ?