Permutation under Restriction

Q. The total number of four-digit odd numbers that can be formed using 0, 1, 2, 3, 5, and 7 (repetition allowed) is

1. 216
2. 375
3. 400
4. 720

Q.

Find the number of ways in which $5$ boys and $5$ girls may be seated in a row so that no two girls and no two boys are together.

Q.

$12$ boys and $2$ girls are to be seated in a row such that there are at least $3$ boys between the $2$ girls. The number of ways this can be done is $m\times 12!$ where $m=?$

1. $2\times {}^{12}C_6$

2. $20$

3. ${}^{11}P_2$

4. ${}^{11}C_2$

Q.

A cake is cut into two equal parts and each part is again cut into two equal halves and it continues in that way. How many times should one cut the cake so that the number of pieces will be 32? [3 MARKS]

Q. How many three digit can be formed using the digits $1,2,3,4,5,$ without repetition of digits$?$ How many of these are even$?$

Q.

The number of $3$ digit odd numbers, that can be formed by using the digits $1,2,3,4,5,6$ when the repetition is allowed, is

1. $60$

2. $108$

3. $36$

4. $30$

Q. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose the chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. the number of possible arrangements is

1. ${}^6{C}_3{\times }^4{C}_2$
2. ${}^4{P}_2{\times }^4{P}_3$
3. ${}^4{C}_2{+}^4{P}_3$
4. none of these

Q.

Write the face value and place value of each odd digit in the given number:

$32,35,75,896$

Q.

Observe the following pattern and find the missing numbers.

${11}^2=121$

${101}^2=10201$

${10101}^2=102030201$

${1010101}^2=\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_{\_}^2=10203040504030201$

Q. The number of integers greater then 6000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is: -

1. 216
2. 192
3. 120
4. 72

Q. If a denotes the number of permutations of (x+2) things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of (x – 11) things taken all at a times and a = 182bc, then x =

1. 13
   
2. 12
3. 14
4. 15

Q. A letter lock consists of $4$ rings, each ring contains $9$ non-zero digits. This lock can be opened by setting a $4$ digit code with the proper combination of each of the 4 rings. Maximum how many codes can be formed to open the lock?

1. $9^4$
2. $4^9$
3. $7^4$
4. None of these

Q. Evaluate: $\frac{(3\frac{2}{3}{)}^2-(2\frac{1}{2}{)}^2}{(4\frac{3}{2}{)}^2-(3\frac{1}{3}{)}^2}$ $÷$ $\frac{3\frac{2}{3}-2\frac{1}{2}}{4\frac{3}{2}-3\frac{1}{3}}$

1. $\frac{35}{53}$
2. $\frac{42}{59}$
3. $\frac{37}{53}$
4. $\frac{47}{60}$

Q. The number of five digit numbers having at least one of their digits repeated is:

Q. How many different ways can $15$ candy bars be distributed between Ram, Shyam, Ghanshyam and Balram, if Ram can not have more than $5$ candy bars and Shyam must have at least two ? Assume all candy bars to be alike.

Q. How many times will you write even numerals if you write all the numbers from $491$ to $500$.

1. $11$
2. $13$
3. $12$
4. $15$

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

Q.

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue.
Then the value of $\frac{m}{n}$ is ___

Q. In a queue of $28$ students, the position of Neeraj is $12$ from forward, then what will be the position of Neeraj from backward ?

1. $19$
2. $17$
3. $16$
4. $18$