Permutation under Restriction

Q. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Q.

How many numbers can be formed using the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

Q.

Find the number of permutations of 7 objects, taken 3 at a time.

Q. In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together?

Q.

6 women and 5 men are to be seated in a row so that no 2 men can sit together. Number of ways they can be seated is

1. 6! $\times {}^6{P}_5$

2. 5! $\times {}^7{P}_5$

3. 5! $\times {}^7{P}_5$

4. 6! $\times {}^7{P}_5$

Q. How many $3$-digit numbers can be formed by using the digits $1$ to $9$ if no digit is repeated?

Q. Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4\times 4$ matrix given in the ORS.
Column I Column II

(A)The number of permutations containing the word ENDEA is (p)5!
(B)The number of permutations in which the letter # occurs in the first and the last positions is (q) $2\times 5!$
(C)The number of permutations in which none of the lettersD, L, N, occurs in the last five jpositions is (r) $7\times 5!$
(D) The number of pernutations in which the leters A, E, O occur only in odd positions is (s)$21\times 5!$

1. The number of permutations in which none of the lettersD, L, N, occurs in the last five jpositions is (r) $7\times 5!$
2. The number of permutations containing the word ENDEA is (p)5!
3. The number of jpermutations in which the letter # occurs in the first and the last positions is (q) $2\times 5!$
4. The number of pernutations in which the letersA, E, O occur only in odd positions is (s) $21\times 5!$

Q.

A committee of 12 members is to be formed from 9 women and 8 men. In how many ways can this be done, if at least five women have to be included in a committee?

Q. Eight different letters of an alphabet are given. Words of four letters from these are formed. The number of such words with atleast one letter repeated is

1. 
2. 
3. 
4. 

Q.

How many different words can be formed by the letters of the word $MISSISSIPPI$ in which number two are adjacent ?

1. $8\times {}^{6}C_4\times {}^{7}C_4$

2. $2\times 7\times {}^{8}C_4$

3. $6\times 8\times {}^{7}C_4$

4. $7\times {}^{6}C_4\times {}^{8}C_4$

Q. The number of ways in which five students of different heights can sit in a row so that the tallest and the shortest may not sit together is

1. 24
2. 48
3. 14
4. 72

Q. Consider the word ${}^{\prime }TELANGAN{A}^{\prime }$, then the number of possible words

1. without any restrictions is $\frac{9!}{2!3!}$
2. If word starts with consonant is $4\times \frac{8!}{2!3!}$
3. when the word starts and end with any vowel is $7!$
4. If word starts with $T$ and ends with $G$ is $420$

Q. In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

Q. The number of ways in which we can get a sum of $11$ by throwing three dice is :

1. $8$
2. $27$
3. $45$
4. $56$

Q. How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

Q.

Find the number of words with or without meaning which can be made using all the letters of the word SWEET.

1. 120

2. 60

3. 24

4. 48

Q. If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

Q. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Q. The number of ways in which 5 boys and 5 girls can be arranged in a row so that no two girls are together is

1. 10!
2. 5!6!
3. 
4. 

Q. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?

Q. The number of ways in which six + signs and four – signs can be arranged in a row so that no two – sings occur together is

1. 
2. 
3. 
4. 6!7!

Q.

Ten persons, amongst whom are A, B and C to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is

1. 

2. 

3. 

4. None of these

Q. 6 women and 5 men are to be seated in a row so that no 2 men can sit together. Number of ways they can be seated is

1. 5!$\times {}^7{P}_5$
2. 6!$\times {}^6{P}_5$
3. 5!$\times {}^7{P}_5$
4. 6!$\times {}^7{P}_5$

Q.

Describe the sample space for the indicated experiment.
2 boys and 2 girls are in a Room X and 1 boy and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.

Q. How Many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $DAUGHTER$?

Q. How many different numbers of six digits each (without repetition of digits) can be formed from the digits 4, 5, 6, 7, 8, 9?
How many of these are not divisible by 5?

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

Q. In how many ways can the letter of the word $PERMUTATIONS$ can be arranged so that all the vowels come together

Q. A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of:
(i) exactly $3$ girls?
(ii) at least $3$ girls?
(iii) at most $3$ girls?

Q.

Out of 6 boys and 6 girls, a group of 8 is to be formed. In how many ways can this be done if the group should have 4 boys?

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