Permutation: n Different Things Taken All at a Time When All Are Not Different.

Q. How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order ?

1. 120
2. 480
3. 240
4. 360

Q. If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is ?

1. ${46}^{th}$
2. ${52}^{th}$
3. ${59}^{th}$
4. ${58}^{th}$

Q.

How many different nine-digit numbers can be formed from the number $223355888$by rearranging its digits so that the odd digits occupy even positions?

1. $16$

2. $36$

3. $60$

4. $180$

Q. How many $3\times 3$ matrices M with entires from {0, 1, 2} are there, for which the sum of the diagonal entries of $M^{T}M$ is 5 ?

1. 162
2. 198
3. 126
4. 135

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. In a two-digit number, the tens digit is twice the unit digit. If the sum of its digits is $9$, find the number.

Q.

How many different nine- digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

1. 16

2. 36

3. 60

4. 180

Q. How many different nine-digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions ?

1. 16
2. 36
3. 60
4. 180

Q. If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL are arranged in a dictionary. Then the position of the word SMALL is : -

1. $52nd$
2. $58th$
3. $46th$
4. $59th$

Q. How many $3\times 3$ matrices M with entires from {0, 1, 2} are there, for which the sum of the diagonal entries of $M^{T}M$ is 5 ?

1. 198
2. 162
3. 126
4. 135

Q. How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order ?

1. 480
2. 120
3. 360
4. 240

Q. How many positive integers less than $1000$ are $6$ times the sum of their digits ?

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

Q. Find the sum of integers which are divisible by $2$ from $11$ to $50$

Q.

Which of the following collections is not a set?

1. The vowels in the English alphabet

2. All natural numbers which divide 42

3. Names of 10 most renowned actors

4. Names of rivers in India