Question 1: Evaluate each of the following:

i) ii) iii) iv)

Answer:

We know,

i)

ii)

iii)

iv)

Question 2: If , find .

Answer:

Since

Hence

Question 3: If , find .

Answer:

Comparing LHS and RHS we get

Hence

Question 4: If , find .

Answer:

Comparing LHS and RHS we get

Hence

Question 5: If , find the value of .

Answer:

Comparing LHS and RHS we get

Hence

Question 6: If , find

Answer:

Question 7: If find .

Answer:

Comparing LHS and RHS we get

Question 8: If , find .

Answer:

Comparing LHS and RHS we get

Question 9: If , find .

Answer:

Question 10: If find .

Answer:

or (this is not possible)

Question 11: If , find .

Answer:

or ( not possible as cannot be letter than 3)

Question 12: Prove that:

Answer:

LHS

RHS. Hence proved.

Question 13: If , find

Answer:

Comparing LHS with RHS, we get

Question 14: If , find

Answer:

Question 15: In how, many ways can five children stand in a queue?

Answer:

We need to permute children out of .

Therefore the required number of ways

Question 16: From among the teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?

Answer:

We need to permute teachers out of .

Therefore the required number of ways

Question 17: Four letters and , one in each, were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?

Answer:

We need to permute letters out of .

Therefore the required number of ways

Question 18: Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged in a shelf. In how many ways can this be done?

Answer:

We need to permute books out of .

Therefore the required number of ways

Question 19: Find the number of different letter words, with or without meanings, that can be formed from the letters of the word .

Answer:

We need to permute letters out of .

Therefore the required number of ways

Question 20: How many three-digit numbers are there, with distinct digits, with each digit odd?

Answer:

The odd digits are

We need to permute digits out of .

Therefore the required number of ways

Question 21: How many words, with or without meaning, can be formed by using all the letters of the word , using each letter exactly once?

Answer:

Number of letters in are

We need to permute letters out of .

Therefore the required number of ways

Question 22: How, many words, with or without meaning, can be formed by using the letters of the word ?

Answer:

Number of letters in are

We need to permute letters out of .

Therefore the required number of ways

Question 23: There are two works each of volumes and two works each of volumes; In how many ways can the books be placed on a shelf so that the volumes of the same work are not separated?

Answer:

There are 4 different types of works.

Number of ways in which these works can be arranged

Within the works the volumes can be arranged.

Therefore the total number of ways arrangement can be made .

Question 24: There are items in column and items in column . A student is asked to match each item in column with an item in column . How many possible, correct or incorrect, answers are there to this question?

Answer:

There are items in column and items in column .

The first item from column can be matched with any of the items from column .

Similarly, once the first items from column has been matched, the 2nd item from the column can only be matched with items of the column . and so on….

Hence the number of ways the match can be done

The other way to look at this is to keep column fixed and arrange 6 items from column which would be

Question 25: How many three-digit numbers are there, with no digit repeated?

Answer:

Total number of 3 digit number

Total number of 3 digit numbers starting with

Hence the three-digit numbers are there, with no digit repeated

Question 26: How many digit telephone numbers can be constructed with digits if each number starts with and no digit appears more than once?

Answer:

In total there are digits to chose from.

However, the first two digits are already fixed to .

Therefore we can now chose numbers from in

ways.

Therefore , digit telephone numbers can be constructed with digits if each number starts with and no digit appears more than once.

Question 27: In how many ways can boys and girls be arranged for a group photograph if the girls are to sit on chairs in a row, and the boys are to stand in a row behind them?

Answer:

Number of ways you can arrange the girls

Number of ways you can arrange the boys

Hence the total number of arrangements

Question 28: If denotes the number of permutations of things taken all at a time, the number of permutations of things taken at a time and the number of permutations of things taken all at a time such that , find the value of .

Answer:

Given

Comparing LHS with RHS we get

Question 29: How many digit numbers can be formed by using the digits to if no digit is repeated?

Answer:

The total number of 3 digit number that can be formed

Question 30: How many 3- $digit even numbers can be made using the digits if no digits is repeated?

Answer:

The even digits are 2, 4, 6

The number of 3-digit numbers when the last digit is

The number of 3-digit numbers when the last digit is

The number of 3-digit numbers when the last digit is

Hence the total number of 3 digit even numbers that can be formed

Question 31: Find the numbers of digit numbers that can be formed using the digits if no digit is repeated? How many of these will be even?

Answer:

The number of 4 digit numbers that can be formed

The even digits are

The number of 4-digit numbers when the last digit is

The number of 4-digit numbers when the last digit is

Therefore the total number of even numbers

Question 32: All the letters of the word are arranged in different possible ways. Find the number of arrangements in which no two vowels are adjacent to each other.

Answer:

Number of consonants

Number of vowels

No two vowels can be together.

Number of ways the vowels can be arranged

Number of ways consonants can be arranged

Hence the total number of arrangements in which no two vowels are adjacent to each other