Question 1: How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?

Answer:

vowels and consonants can be formed with vowels and consonants can be chosen in

The five letters can be arranged in ways.

Hence the total number of words that can be formed

Question 2: There are 10 persons named Out of 10 persons, 5 persons are to be arranged in a line such that is each arrangement must occur whereas and do not occur. Find the number of such possible arrangements.

Answer:

We have to arrange persons out of persons in such a way that is always there and and are not selected.

So basically we have to select persons out of persons. Hence the number of combinations

Once we have the persons selected, they can be arranged in ways.

Therefore the number of ways persons can be arranged in a line such that is each arrangement must occur whereas and do not occur out of given persons

Question 3: How many words, with or without meaning can be formed from the letters of 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 letter is a vowel?

Answer:

i) Number of 4 lettered words that can be formed from the letters of the word , assuming that no letter is repeated and if 4 letters are used at a time

ii) Number of 4 lettered words that can be formed from the letters of the word , assuming that no letter is repeated and all letters are used at a time

iii) The first place can be filled in 2 ways ( either ). Therefore one vowel can be chosen from 2 vowels in ways.

Remaining 5 letters can be chosen in way

Therefore the number of words that can be formed

Question 4: Find the number of permutations of distinct things taken together, in which 3 particular things must occur together.

Answer:

The number of combination of distinct things taken r together

If three things come together

Number of arrangements of three things

Number of arrangements of objects

So, total possible ways

Question 5: How many words each of 3 vowels and 2 consonants can be formed from the letters of the word ?

Answer:

Given word

Number of letters

Vowels

Consonants

Number of ways to select vowels

Number of ways to select consonants

Therefore the number of ways to arrange these letters

Question 6: Find the number of permutations of different things taken at a time such that specified things occur together

Answer:

The number of combination of distinct things taken together

If three things come together

Number of arrangements of two things

Number of arrangements of objects

So, total possible ways

Question 7: Find the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word .

Answer:

(a) Given word: .

Total number of letters

Number of Number of Number of

Number of Number of Number of

The four letter words may consist of

i) 3 alike letters and distinct letter

Number of ways to select these letters

ii) alike letters of one kind and alike letters of second kind

There are three pairs of letters where there are more than one letters. We need to select any of the letters.

Number of ways to select these letters

iii) alike letters and distinct letters

Number of ways to select these letters

iv) all different letters

Number of ways to select these letters

Therefore the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word

(b) 4 letter word may consist of:

i) 3 alike letters and distinct letter

Number of arrangements with 3 alike and one distinct

ii) alike letters of one kind and alike letters of second kind

Number of arrangements with alike letters of one kind and alike letters of second kind

iii) alike letters and distinct letters

Number of arrangements with two letter are same and 2 are of different kind

iv) all different letters

Number of arrangements for 4 distinct letters

Therefore the total number of arrangements possible

Question 8: How many words can be formed by taking 4 letters at a time from the letters of the word ?

Answer:

Given word: .

Total number of letters

Number of Number of Number of

Number of Number of Number of

i) 3 alike letters and distinct letter

Number of arrangements with 3 alike and one distinct

ii) alike letters of one kind and alike letters of second kind

Number of arrangements with alike letters of one kind and alike letters of second kind

iii) alike letters and distinct letters

Number of arrangements with two letter are same and 2 are of different kind

iv) all different letters

Number of arrangements for 4 distinct letters

Therefore the total number of arrangements possible

Question 9: A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests?

Answer:

15 people can be accommodated on the table in

The remaining 6 people can be arranged in

Hence the total number of ways people can be arranged

Question 10: Find the number of combinations and permutations of 4 letters taken from the word .

Answer:

Given word: .

Total number of letters

Number of Number of Number of

All other letters are not repeated.

i) alike letters of one kind and alike letters of second kind

Number of arrangements with alike letters of one kind and alike letters of second kind

ii) alike letters and distinct letters

Number of arrangements with two letter are same and 2 are of different kind

iii) all different letters

Number of arrangements for 4 distinct letters

Therefore the total number of arrangements possible

Question 11: A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated?

Answer:

We have to arranged for persons along two sides of a long table with chairs on each side.

persons wish to sit on one particular side and on the other side .

Out of the people left, people can be seated on side in ways.

And from the remaining people can be selected for side in ways.

Hence the number of selections

Now people on each side can be arranged in ways.

Therefore the total number of ways in which the people can be seated