Question 1: Find the number of words formed by permuting all the letters of the following words:
Answer:
There are letters in the word
out of which
,
and
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
,
and
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
,
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
,
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
,
and the rest are all distinct.
Therefore the total number of words
There are letters in the word
out of which
,
and
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
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
and the rest are all distinct.
There are vowels in the word
out of which
and the rest are all distinct.
ways
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
,
,
.
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
,
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
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
and the rest are all distinct.
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
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 numbers of four digits can be formed with the digits ?
Answer:
Total number of digits
However, cannot be in the first place.
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
,
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
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
.
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 on a shelf?
Answer:
There are three copies each of different books
Total number of books
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
,
,
and the rest are all distinct.
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.
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
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.
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
,
,
,
and the rest are all distinct.
Considering all together, we need to arrange 10 letters.
Question 21: Find the total number of permutations of the letters of the word ?
Answer:
There are letters in the word
out of which
,
and the rest are all distinct.
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
If you arrange them in alphabetical order then we have
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
( 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
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
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 does not alter?
Answer:
There are letters in the word
out of which
,
,
and the rest are all distinct.
ii) The relative order of vowels and consonants do not alter.
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
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 ,