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

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.

The letters of the word SACHIN are permuted and are arranged in an alphabetical order as in an English dictionary. Then, the rank of the word SACHIN is _____

___

Q.

How many words (with or without meaning) can be formed from the letters of the word, 'DAUGHTER', so that

(i) all vowels occur together?

(ii) all vowels do not occur together?

Q.

Find the degree measures corresponding to the following radian measures $(\text{use}\pi =\frac{22}{7})$:

(i) $\frac{11}{16}$ (ii) -4 (iii) $\frac{5\pi }{3}$ (iv) $\frac{7\pi }{6}$

Q.

Find the sum of n terms of the series 0.3 + 0.33 + 0.333 + . . . .

Q. Number of ways of permuting the letters of the word “ENGINEERING” so that the order of the vowels is not
changed is

1. 
2. 
3. 
4. 

Q.

How many odd numbers less than 1000 can be formed by using the digits 0, 2, 5, 7 when the repetition of digits is allowed ?

Q.

The last three digits of the number $(81{)}^{25}$ are x, y and z. Find x+y+z

__

Q.

A person writes a letter of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

Q. Find the value of sgn (-1) + sgn (1) + sgn (5) + sgn (0), where sgn is the signum function.
___

Q.

Words are made with or without meaning from letters of the word AGAIN. If these words are written as in a dictionary , what is the ${55}^{th}$ word?

1. INAAG

2. NGAAI

3. GIANA

4. ANAGI

Q.

How many words, with or without meaning can be formed from the letters of the word 'MONDAY' assuming that no letter is repeated? If

(i) 4 letters are used at a time?

(ii) all letters are used at a time ?

(iii) all letters are used but first letter is vowel?

Or

How many different words can be formed by using all the letters of the word 'ALLAHABAD' ?

(i) In how many of them, vowels occupy the even position ?

(ii) In how many of them, both 'L' do not come together ?

Q.

Write the set $G=\{1,3,5,7,9,11,\dots \}$ in the set-builder form.

Q.

Write the first five terms of the sequences whose $n^{th}$ term is :

$a_n=n\frac{n^2+5}{4}$

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. 180

4. 60

Q.

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

Q. Are the following pair of sets equal ? Give reasons.
(i) $A=\{2,3\},B=\{x:x\text{is solution of}{x}^2+5x+6=0\}$
(ii) $A=\{x:x\text{is a letter in the word FOLLOW}\}$

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. 360
4. 240

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.

Write the first five terms of the sequences whose $n^{th}$ term is:

$a_n=\frac{2n-3}{6}$

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

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