Question 1: Find the number of words formed by permuting all the letters of the following words:

i) ii) iii) iv) v) vi) vii) viii) ix)

Answer:

i)

There are letters in the word out of which are , are and are and the rest are all distinct.

Therefore the total number of words

ii)

There are letters in the word out of which are , are and are and the rest are all distinct.

Therefore the total number of words

iii)

There are letters in the word out of which are , are and the rest are all distinct.

Therefore the total number of words

iv)

There are letters in the word out of which are and the rest are all distinct.

Therefore the total number of words

v)

There are letters in the word out of which are and the rest are all distinct.

Therefore the total number of words

vi)

There are letters in the word out of which are and the rest are all distinct.

Therefore the total number of words

vii)

There are letters in the word out of which are , are and the rest are all distinct.

Therefore the total number of words

viii)

There are letters in the word out of which are , are and the rest are all distinct.

Therefore the total number of words

ix)

There are letters in the word out of which are , are and are and the rest are all distinct.

Therefore the total number of words

Question 2: In how many ways can the letters of the word be arranged without changing the relative order of the vowels and consonants?

Answer:

Number of consonants in

Number of vowels in of which are

Therefore the number of ways you can arrange the consonants

And the number of ways you can arrange the vowels

Hence the total number of ways the letters of the word be arranged without changing the relative order of the vowels and consonants

Question 3: How many words can be formed with the letters of the word , the vowels remaining together?

Answer:

There are letters in the word out of which are and the rest are all distinct.

There are vowels in the word out of which are and the rest are all distinct.

Considering all vowels as one, letters can be arranged in ways

The vowels themselves can be arranged in ways.

Hence the number of words can be formed with the letters of the word , the vowels remaining together

Question 4: Find the total number of arrangements of the letters in the expression when written at full length.

Answer:

There are letters in the in the expression out of which are , are , are .

Therefore the total number of expressions

Question 5: How many words can be formed with the letters of the word so that all do not come together?

Answer:

There are letters in the word out of which are , are and the rest are all distinct.

Therefore the total number of words possible

Now let us find the number of words where are together.

Considering all as one, letters, including are can be arranged in

Therefore, the number of words that can be formed with the letters of the word so that all do not come together

Question 6: How many words can be formed by arranging the letters of the word so that all come together?

Answer:

There are letters in the word out of which are and the rest are all distinct.

Considering all as one, letters can be arranged in ways

Question 7: How many numbers can be formed with the digits so that the odd digits always occupy the odd places?

Answer:

Total number of digits

Number of odd digits

Number of odd places

The odd digits can be arranged in

The rest of the three even digits 2,2,4 can be arranged in

Hence the total number of arrangements

Question 8: How many different signals can be made from red, white and green flags by arranging all of them vertically on a flagstaff?

Answer:

Total number of flags

Number of red flags

Number of white flags

Number of green flags

Total number of signals that can be created

Question 9: How many number of four digits can be formed with the digits ?

Answer:

Total number of digits

Number of digit numbers that can be formed

However, cannot be in the first place.

Number of three digit numbers that can be formed

Therefore the number of four digits can be formed with the digits

Question 10: In how many ways can the letters of the word be arranged so that the two are never together?

Answer:

There are letters in the word out of which are , are and the rest are all distinct.

Therefore the total number of words possible

Now let us find the number of words where are together.

Considering all as one and we have are can be arranged in

Therefore, the number of words that can be formed with the letters of the word so that all do not come together

Question 11: How many different numbers, greater than can be formed with the digits .

Answer:

Total number of digits

For the number to be greater then , the first digit should be either or .

Number arrangements with in the first place

Number arrangements with in the first place

Therefore the different numbers, greater than can be formed with the digits

Question 12: How many words can be formed from the letters of the word which start with and end with ?

Answer:

We have to fill in the middle 4 boxes.

Therefore the number of words that can be formed from the letters of the word which start with and end with

Question 13: How many permutations of the letters of the word do not begin with but end with ?

Answer:

Total number of letters in

Total number of words that can be made from ending with

Number of word starting with and ends with

Therefore the number of words that do not begin with but end with

Question 14: Find the number of numbers, greater than a million, that can be formed with the digits .

Answer:

Total number of digits

Total number of numbers that can be made from digits

If is at the first position. Total number of numbers that can be made from digits

Hence the total number of numbers, greater than a million, that can be formed with the digits

Question 15: There are three copies each of different books. In how many ways can they be arranged in a shelf?

Answer:

There are three copies each of different books

Total number of books

Hence the number of possible arrangements

Question 16: How many different arrangements can be made by using all the letters in the word . How many of them begin with ? How many of them begin with ?

Answer:

There are letters in the word out of which are , are , are and the rest are all distinct.

Therefore the number of words that can be made

If is the first character, then the number of words that can be made

If is the first character, then the number of words that can be made

Question 17: A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?

Answer:

Total number of molecules

The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind.

Therefore the total number of possible arrangements

Question 18: In how many ways can red, yellow and green discs be arranged in a row if the discs of the same color are indistinguishable?

Answer:

Total number of discs

We have red, yellow and green discs

Therefore the total number of possible arrangements

Question 19: How many numbers greater than can be formed by using the digits ?

Answer:

Total number of digits

Total number of numbers that can be made from digits

If is at the first position.

Total number of numbers that can be made from digits

Hence the total number of numbers, greater than a 1000000, that can be formed with the digits

Question 20: In how many ways can the letters of the word be arranged so that all the are together?

Answer:

There are letters in the word out of which are , are , are , are and the rest are all distinct.

Considering all together, we need to arrange 10 letters.

Hence the 10 letters can be arranges in

Question 21: Find the total number of permutations of the letters of the word ?

Answer:

There are letters in the word out of which are , are and the rest are all distinct.

Total number of permutations of the letters of the word

Question 22: The letters of the word are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word .

Answer:

In a dictionary, the words are arranged in alphabetical order.

Distinct letters in are

If you arrange them in alphabetical order then we have

Number of words that start with

Number of words that start with

Now we need to find words that start with S and are before

These would be words starting with .

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with ( this includes ) This will also contain which will be after in the dictionary. Hence you have to subtract it.

Number of words that start with ( this includes )

Hence the rank of

Question 23: If the letters of the word be permuted and the words so formed be arranged as in a dictionary, find the rank of the word .

Answer:

In a dictionary, the words are arranged in alphabetical order.

Distinct letters in are

If you arrange them in alphabetical order then we have

Number of words that start with

Number of words that start with

Number of words that start with But one of the words itself is

Hence the rank

Question 24: If the letters of the word are written in all possible orders and these words are written out as in a dictionary, find the rank of the word .

Answer:

In a dictionary, the words are arranged in alphabetical order.

Distinct letters in are

If you arrange them in alphabetical order then we have

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with However, in this combination, there will be the following words MOTHER, MOTHRE, MOTEHR, MOTERH, MOTRHE, MOTREH. Of which one is MOTHER and MOTHRE, MOTRHE, MOTREH will come after MOTHER. Hence the total number to count there is only .

Hence the rank of MOTHER is

Question 25: If the permutations oi taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation .

Answer:

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with that we need to consider =

Therefore the number of words after which you will reach

Question 26: Find the total number of ways in which six and four signs can be arranged in a line such that no two signs occur together.

Answer:

Number of sign

Number of sign

signs can be arranged in way.

Between the signs, there are places where sign can go.

Therefore the identical signs can be arranged in

Hence the total number of ways to arrange the signs

Question 27: In how many ways can the letters of the word be arranged so that:

(i) the vowels always occupy even places?

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

Answer:

There are letters in the word out of which are , are , are and the rest are all distinct.

i) Six vowels can be arranged in six even places

The remaining six consonants can be arranged in

Therefore the total possible arrangements

ii) The relative order of vowels and consonants do not alter.

Arranging six vowels without disturbing their relative places

Similarly, arranging six consonants without disturbing their relative places

Therefore the total possible arrangements

Question 28: The letters of the word are written in all possible orders. How many words are possible if all these words are written out as in a dictionary ? What is the rank of the word , ?

Answer:

In a dictionary, the words are arranged in alphabetical order.

Distinct letters in are

If you arrange them in alphabetical order then we have

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Number of words that start with

Therefore the rank of the word ,