Permutation under Restriction

Q.

What are ${}^{n}P_r$ and ${}^{n}C_r$ in math?

Q.

What is it that comes once in a year, twice in a month, thrice in a week and four times in a day?

Q. The number of ways in which $100$ people can be divided in $50$ pairs is

1. $\frac{100!}{2^{50}(50!)}$
2. $\frac{100!}{2^{50}}$
3. $\prod _{n=26}^{50}\frac{{}^{2n}{C}_2}{2}$
4. $\prod _{n=1}^{50}{}^{2n}{C}_2$

Q. The number of $4$ digit numbers formed by $0,1,2,3,4,5$ (repetition of digits is allowed) such that it is divisible by $6$ is.

Q. A coin is tossed $3$ times and the outcomes are recorded. How many possible outcomes are there?

Q.

The number of words from the letters of the word 'Bharat' in which B and H will never come together, is

1. $360$

2. $120$

3. $240$

4. None of these

Q.

How many permutations can be formed by the letters of the word, 'VOWELS', when
(i) there is no restriction on letters?
(ii) each word begins with E?
(iii) each word begins with O and ends with L?
(iv) all vowels come together?
(v) all consonants come together?

Q. The number of natural numbers less than $7,000$ which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to :

1. $372$
2. $375$
3. $374$
4. $250$

Q.

$18$ mice were placed in two experimental groups and one control group, with all group equally large. In how many ways can the mice be placed into three groups ?

Q. The number of $6$ digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by $11$ and no digit is repeated, is :

1. $48$
2. $60$
3. $72$
4. $36$

Q.

All the words that can be formed using alphabets $A,H,L,U,$ and $R$ are written as in a dictionary (no alphabet is repeated). The rank of the word $RAHUL$ is

1. $71$

2. $72$

3. $73$

4. $74$

Q.

How many odd numbers less than 1000 can be formed by using the digits 0, 3, 5, 7 when repetition of digits is not allowed ?

Q.

In how many ways can the letters of the word 'STRANGE' be arranged so that
(i) the vowels come together?
(ii) the vowels never come together? and
(iii) the vowels occupy only the odd places?

Q. The number of numbers between $2000$ and $5000$ that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of $3$ is :

1. $36$
2. $30$
3. $24$
4. $48$

Q.

Number of $5$ digit numbers which are divisible by $5$ and each number containing the digit $5$, digits being all different is equal to $K\left(4!\right)$, then the value of $K$ is

1. $84$

2. $168$

3. $188$

4. $208$

Q.

How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?

Q.

The number of four-letter words that can be formed (the words need not be meaningful) using the letters of the word MEDITERRANEAN’ such that the first letter is $E$ and the last letter is $R$, is

1. $\frac{11!}{2!·2!·2!}$

2. $59$

3. $56$

4. $\frac{11!}{3!·2!·2!}$

5. $\frac{11!}{3!·3!·2!}$

Q. The letters of word $OUGHT$ are written in all possible orders and these words are written out as in a dictionary. Then the rank of the word $TOUGH$ is

1. $90$
2. $91$
3. $92$
4. $89$

Q.

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_____.

Q. Let the equation $x^2+y^2+px+(1–p)y+5=0$ represent circles of varying radius $r\in (0,5]$. Then the number of elements in the set $S=\{q:q=p^2$ and $q$ is an integer} is .

Q.

The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is

1. $60\times 5!$

2. $15\times 4!\times 5!$

3. $4!\times 5!$

4. None of these

Q. Number of $4$ letter words starting with vowel that can be formed using letters of the word ${}^{\prime }CHEMISTR{Y}^{\prime }$ are

1. $336$
2. $504$
3. $672$
4. $1008$

Q. Number of five digit numbers that contain digit $7$ exactly once (repetition of digits is allowed) is

1. $41\times 9^3$
2. $37\times 9^3$
3. $41\times 9^4$
4. $7\times 9^4$

Q.

How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?

Q.

In how many ways can the letters of the word 'INTERMEDIATE' be arranged so that :

(i) the vowels always occupy even places ?

(ii) the relative order of vowels and consonants do not alter ?

Q.

The number of positive integers which can be formed by using any number of digits from $0,1,2,3,4,5$but using each digit not more than once in each number, is equal to:

1. $1200$

2. $1500$

3. $1600$

4. $1630$

Q.

All the letters of the word 'EAMCOT' are arranged in different possible ways. find the number of arrangement in which no two vowels are adjacent to each other.

Q. The letters of the word "RANDOM" are arranged in all possible ways. The number of arrangements in which there are $2$ letters between 'R' and 'D' is

1. $36$
2. $48$
3. $72$
4. $144$

Q. Let $N=2^4\times 3^5\times 5^2\times 7^4\times 11$ be resolved as a product of two factors (order immaterial), then

1. number of cases in which the two factors are coprime is $16$
2. number of cases in which their H.C.F. is $3$, is $32$
3. number of cases in which their H.C.F. $5$, is $16$
4. number of cases in which their H.C.F is odd is $180$

Q.

In how many ways$n$ books can be arranged in a row so that two specified books are not together?

1. $n!–(n–2)!$

2. $(n–1)!–(n–2)$

3. $n!–2(n–1)!$

4. $(n–2)n!$