Question 1: In how many ways can the letter so the word, , be arranged so that the consonants may occupy only odd positions?

Answer:

Number of letters in

There are 4 vowels and 3 consonants in . In all we have 7 places to be filled.

Consonants can go to boxes in ways

The 4 vowels can be arranged in the remaining places in ways.

Hence the total number of arrangements

Question 2: In how many ways can the letters of the word be arranged so that:

i) the vowels come together ? ii) the vowels never come together? and iii) the vowels occupy only the odd places?

Answer:

Number of letters in = 7

Number of vowels

Number of consonants

i) Considering two vowels as one, we can arrange the in ways.

However, the vowels themselves can be arranges in ways.

Hence the total number of ways the letters can be arranges ways.

ii) Total number of ways that all the letter can be arranged ways.

Therefore the total number of words where the vowels never come together

iii)

There are odd places. The two vowels can be arranged in ways.

The rest of the consonants can be arranged in ways.

Hence the total number of arrangements

Question 3: How many words can be formed from the letters of the word ? How many of these begin with ?

Answer:

Number of letters in

Therefore the total number of words that can be formed

The total number of words that can be formed with D fixed at the first position

Question 4: How many words can be formed out of the letters of the word, , so that the vowels , always occupy the odd places ?

Answer:

Number of letters in

There are 4 vowels in .

4 vowels can go to boxes in ways

The 4 consonants can be arranged in the remaining places in ways.

Hence the total number of arrangements

Question 5: How many words can be formed out of the letters of the word ? How many of these words begin with ? How many begin with and end with .

Answer:

Number of letters in

Total number of words that can be formed

Total number of words that can be formed with in the first position

Total number of words that can be formed with in the first position and in the last position

Question 6: How many different words can be formed from the letters of the word ? In how many of these words:

(i) the letter always occupies the first place?

(ii) the letters and respectively occupy first and last place?

iii) the vowels are always together

(iv) the vowels always occupy even places?

Answer:

Number of letters in

i) Total number of words that can be formed with in the first position

ii) Total number of words that can be formed with in the first position and in the last position

iii) There are vowels and consonants. Considering all vowels as one, we can arrange the in

The 4 vowels can themselves be arranged in ways.

Hence the total possible arrangements

iv)

There are even places . Therefore vowels can be arranged in

Remaining places can be occupies by the consonants in ways.

Hence the total number of arrangements

Question 7: How many permutations can be formed by the letters of the word when

i) there is no restriction on letters ii) each word begins with letter

iii) each word begins with and ends with iv) all vowels come together

v) all consonants come together

Answer:

Number of letters in

i) Total number of words that can be formed

ii) Total number of words that can be formed with in the first position

iii) Total number of words that can be formed with in the first position and in the last position

iv) There are vowels and consonants. Considering all vowels as one, we can arrange the in

The vowels can themselves be arranged in ways.

Hence the total possible arrangements

v) There are vowels and consonants. Considering all consonants as one, we can arrange the in

The consonants can themselves be arranged in ways.

Hence the total possible arrangements

Question 8: How many words can be formed out of letters of the word , so that vowels occupy even places.

Answer:

Number of letters in

Number of vowels

There are even places . Therefore vowels can be arranged in

Remaining places can be occupies by the consonants in ways.

Hence the total number of arrangements

Question 9: In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?

Answer:

We can pick husbands in ways.

Now we cannot pick the two wives for the mixed doubled. We have to choose wives from the left in ways.

Hence the total number of ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set

Question 10: men and women are to be seated in a row so that no two women sit together. If then show that the number of ways in which they can be seated as

Answer:

men can be seated in ways.

Once the men are seated, there would be spots for the women to be seated in. Therefore the number of ways women can be seated in ways.

Hence the total number of ways men and women are to be seated in a row so that no two women sit together

Hence proved.

Question 11: How ‘many words (with or without dictionary meaning) can be made from the letters in the word , 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 is vowel?

Answer:

i) Number of letters words that can be made

ii) Number of letters words that can be made

iii) First letter is a vowel. There are two vowels.

Hence the number of words that can be made

Question 12: How many three letter words can be made using the letters of the word ?

Answer:

Number of letters in

Number of letters words that can be made