Question 1: In a class there are boys and girls. The teacher wants to select boy and girl to represent the class in a function. In how many ways can the teacher make this selection?

Answer:

Number of boys in the class

Therefore number of ways to select a boy

Number of girls in the class

Therefore number of ways to select a girl

Therefore the number of ways to select boy and girl to represent the class in a function .

Question 2: A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are fountain pen varieties, ball pen varieties and pencil varieties, in how, many ways can he select these articles?

Answer:

Number of ways fountain pen varieties

Therefore number of ways to select a fountain pen

Number of ways ball pen varieties

Therefore number of ways to select a ball pen

Number of ways pencil varieties

Therefore number of ways to select a pencil

Hence the total number of ways to select a fountain pen, a ball pen and a pencil

Question 3: From Goa to Bombay there are roots; air, and sea. From Bombay to Delhi there are routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there?

Answer:

Number of routes from Goa to Bombay

Number of routes from Bombay to Delhi

Therefore the number of routes from Goa to Delhi via Bombay

Question 4: A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?

Answer:

There are two types of years that the mint needs to take care. Non-leap year and leap year.

Each year can start with any of the days.

Hence, the mint needs to prepare types of calendars to serve for all the possibilities in future years

Question 5: There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?

Answer:

Total number of parcels

Total number of post offices

Since a parcel can be sent by registered post by any of the post office, the number of different ways can the parcels be sent by registered post

Question 6: A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?

Answer:

There are possible outcomes when a coin is tossed.

If the coin is tossed 5 times, then the possible outcomes

Question 7: In how many ways can an examinee answers a set of ten true / false type questions?

Answer:

There are two possible ways to answer any question i.e. or

Therefore the total possible ways to answer all questions

Question 8: A letter lock consists of rings each marked with different letters. In how many it is possible to make an unsuccessful attempt to open the lock?

Answer:

The number of possible setting on each of the ring

Number of rings

Therefore the total number of combinations

However, there is combination that is correct. Hence the number of possible unsuccessful attempts

Question 9: There are multiple choice questions in an examination. How many sequences of answers are possible, if the first questions have choices each and the next have each?

Answer:

Number of ways you could answer the first questions

Number of ways you could answer the next questions

Therefore the total possible ways of answering the questions

Question 10: There are books on Mathematics and books on Physics in a book shop. In how many ways can a student buy : (i) a Mathematics book and a physics ii) either a Mathematics book or a Physics book?

Answer:

Number of mathematics books

Therefore the number of ways of buying a Mathematics book

Number of Physics books

Therefore the number of ways of buying a Physics book

i) The number of ways of buying a Mathematics book and a Physics

ii) The number of ways of buying a Mathematics book or a Physics book

Question 11: Given flags of different colors, how many different signals can be generated if a signal requires the use of two flags, one below the other?

Answer:

Number of flags

Number of ways of selecting the first flag

Number of ways of selecting the second flag

Therefore the number of different signals can be generated if a signal requires the use of two flags, one below the other

Question 12: A team consists of boys and girls and other has boys and girls. How many single matches can be arranged between the two teams when a boy plays against a boy and a girl plays against a girl?

Answer:

Number of ways to select a boy from first team

Number of ways to select a boy from second team

Therefore the number of ways of arranging singles matches between boys of the first team and the second team

Number of ways to select a girl from first team

Number of ways to select a girl from second team

Therefore the number of ways of arranging singles matches between girls of the first team and the second team

Therefore the total number of singles matches that can be arranged

Question 13: students compete in a race. In how many ways can the first three prizes can be given.

Answer:

The first prize can be given in ways.

The second prize can be given in ways.

The third prize can be given in ways.

Hence the first three prizes ways

Question 14: How many A.P.’s with 10 terms are there whose first term is in the set and whose common difference is in the set ?

Answer:

Number of ways of selecting the first term

For each of the first terms, there can be common differences that can be chosen.

Hence the number of A.P’s would be

Question 15: From among the teachers in a college, one principal, one vice-principal and the teacher-in charge are to be appointed. In how many ways can this be done?

Answer:

Number of teachers in the college

Number of ways of selecting a Principal

Number of ways of selecting a Vice-Principal

Number of ways of selecting a teacher in-charge

Hence the number of ways this can be done

Question 16: How many three-digit numbers are there with no digit repeated?

Answer:

We have to form all possible digit numbers where no digit is repeated.

A three digit number has a space, space and a units place.

In all there are digits

Hence the space can be filled in possible ways. We cannot have in the space for the number to be a three digit number.

The space can be filled in different ways.

The units place can be filled in different ways.

Therefore the number of three-digit numbers are there with no digit repeated

Question 17: How many three-digit numbers are there?

Answer:

We have to form all possible digit numbers.

A three digit number has a space, space and a units place.

In all there are digits

Hence the space can be filled in possible ways. We cannot have in the space for the number to be a three digit number.

The space can be filled in different ways. The digits are allowed to repeat.

The units place can be filled in different ways. The digits are allowed to repeat.

Therefore the number of three-digit numbers are there with no digit repeated

Question 18: How many three-digit odd numbers are there?

Answer:

We have to form all possible digit odd numbers.

A three digit number has a space, space and a units place.

In all there are digits

Hence the space can be filled in possible ways. We cannot have in the space for the number to be a three digit number.

The space can be filled in different ways. The digits are allowed to repeat.

The units place can be filled in different ways. The digits can be from for the number to be odd.

Therefore the number of three-digit numbers are there with no digit repeated

Question 19: How many different five-digit number license plates can be made if

i) first digit cannot be zero and the repetition of digits is not allowed,

ii) the first-digit cannot be zero, but the repetition of digits is allowed?

Answer:

i) The first digit can be filled in different ways. cannot be considered.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

Five-digit number license plates can be made if the first digit cannot be zero and the repetition of digits is not allowed

ii) The first digit can be filled in different ways. cannot be considered.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

Five-digit number license plates can be made if the first-digit cannot be zero, but the repetition of digits is allowed

Question 20: How many four-digit numbers can be formed with the digits which are greater than , if repetition of digits is not allowed?

Answer:

The first digit can be filled in ways. Only or can go there as the number should be greater than .

The second digit can be filled in different ways. Remember, digits cannot be repeated.

The third digit can be filled in different ways. Remember, digits cannot be repeated.

The fourth digit can be filled in way. Remember, digits cannot be repeated.

Therefore the four-digit numbers that can be formed with the digits which are greater than with no repetition of digits

Question 21: How many four digits numbers can be formed with the digits which are greater than , if repetition of digits is not allowed?

Answer:

The first digit can be filled in ways. Only or can go there as the number should be greater than .

The second digit can be filled in different ways. Remember, digits cannot be repeated.

The third digit can be filled in different ways. Remember, digits cannot be repeated.

The fourth digit can be filled in way. Remember, digits cannot be repeated.

Therefore the four-digit numbers that can be formed with the digits which are greater than with no repetition of digits

Question 22: ln how many ways can persons be seated in a row?

Answer:

The first seat can be filled in ways.

The second seat can be filled in ways.

The third seat can be filled in ways.

The fourth seat can be filled in ways.

The fifth seat can be filled in ways.

The sixth seat can be filled in ways.

Therefore the number of ways persons be seated in a row

Question 23: How many 9-digit numbers of different digits can be formed?

Answer:

Note: Repetition is not allowed as ‘9-digit numbers of different digits’

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

The sixth digit can be filled in different ways.

The seventh digit can be filled in different ways.

The eight digit can be filled in different ways.

The ninth digit can be filled in different ways.

Therefore the number of 9-digit numbers of different digits can be formed or

Question 24: How many odd numbers less than can be formed by using the digits when repetition of digits is not allowed?

Answer:

Note: Repetition is not allowed. Also for number to be less than , it can be a three digit number, two digit number and a single digit number.

Three digit number

The first digit can be filled in different ways. We can only select from

The second digit can be filled in different ways.

The third digit can be filled in different ways. We cannot have in the last digit as the number is odd.

Therefore the number of three digits odd numbers less than can be formed by using the digits when repetition of digits is not allowed

Two digit number

The first digit can be filled in different ways. We can only select from .

The second digit can be filled in different ways. No repetition is allowed and cannot be considered as the number needs to be odd.

Therefore the number of two digits odd numbers less than can be formed by using the digits when repetition of digits is not allowed

Single digit number

The unit digit can be filled in different ways.

Therefore the total number of numbers which are odd and less than that can be formed from

Question 25: How many 3-digit numbers are there, with distinct digits, with each digit odd?

Answer:

The odd digits are . There is no repetition allowed ‘distinct digits’.

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

Therefore the number of 3-digit numbers are there, with distinct digits, with each digit odd

Question 26: How many different numbers of six digits each can be formed from the digits when repletion of digits is not allowed.

Answer:

Note: Repetition is not allowed.

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

The sixth digit can be filled in different ways.

Therefore the numbers of six digits each can be formed from the digits when repletion of digits is not allowed

Question 27: How many different numbers of six digits each can be formed from the digits when repletion of digits is not allowed.

Answer:

Note: Repetition is not allowed.

The first digit can be filled in different ways.

The second digit can be filled in different ways. is not allowed in the sixth digit.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

The sixth digit can be filled in different ways.

Therefore the numbers of six digits each can be formed from the digits when repletion of digits is not allowed

Question 28: How many four digit different numbers, greater than can be formed with the digits when repetition of digits is not allowed?

Answer:

The first digit can be filled in different ways. Only 5 and 9 can be considered.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

Therefore the four digit different numbers, greater than can be formed with the digits when repetition of digits is not allowed

Question 29: Serial numbers for an item produced in a factory are to be made using two letters followed by four digits . If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in numbers a serial number, how many serial are possible?

Answer:

The first digit ( Alphabet) can be filled in different ways.

The second digit ( Alphabet) can be filled in different ways.

The third digit (Number) can be filled in different ways.

The fourth digit (Number) can be filled in different ways.

The fifth digit (Number) can be filled in different ways.

The sixth digit (Number) can be filled in different ways.

Therefore the number of serial numbers that can be created

Question 30: A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats now many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.

Answer:

The number of possible setting on each of the ring

Number of rings

Therefore the total number of combinations

However, there is combination that is correct. Hence the number of possible unsuccessful attempts

Question 31: A customer forgets a four digit code for an Automatic Teller Machine (ATM) in a bank. However, he remembers that this code consists of digits . Find the largest possible number of trials necessary to obtain the correct code.

Answer:

Note: There is no repetition of the digits.

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

Maximum number of code combinations

Question 32: In how many ways can three jobs be assigned to three persons if one person is assigned only one job and all are capable if doing each job?

Answer:

Number of ways can be assigned Jobs = 3

Number of ways can be assigned Jobs = 2

Number of ways can be assigned Jobs = 1

The three jobs be assigned to three persons possible ways.

Question 33: How many four digit natural numbers not exceeding can be formed with the digits and , if the digits can repeat?

Answer:

Four digit natural numbers not exceeding can be digits, digits, digits and single digits.

digit natural numbers not exceeding

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

Therefore the total number four digit different numbers

Now lets calculate the numbers formed greater than $latex 4321.

When the first digit is , then the numbers formed

Also there are are less then $latex 4321.

Hence the total numbers formed less than

digit natural numbers not exceeding

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

Therefore the three digit different numbers

digit natural numbers not exceeding

The first digit can be filled in different ways.

The second digit can be filled in different ways.

Therefore the three digit different numbers

Single digit natural numbers not exceeding

The single digit can be filled in different ways.

Therefore the three digit different numbers

Therefore the total number of number

Question 34: How many numbers of six digits can be formed from the digits 0, 1, 3, 5,7 and 9, when no digit is repeated? How many of them are divisible by 10?

Answer:

The first digit ( Alphabet) can be filled in different ways.

The second digit ( Alphabet) can be filled in different ways.

The third digit (Number) can be filled in different ways.

The fourth digit (Number) can be filled in different ways.

The fifth digit (Number) can be filled in different ways.

The sixth digit (Number) can be filled in different ways.

Therefore the total number of six digit numbers

For the numbers to be divisible by 10, the last digit should be 0. Hence,

The first digit ( Alphabet) can be filled in different ways.

The second digit ( Alphabet) can be filled in different ways.

The third digit (Number) can be filled in different ways.

The fourth digit (Number) can be filled in different ways.

The fifth digit (Number) can be filled in different ways.

The sixth digit (Number) can be filled in different ways.

Therefore the total number of six digit numbers

Question 35: Three six face die each marked with numbers 1 to 6 on six faces, are thrown find , the total number of possible outcomes.

Answer:

The number of possible outcomes on one dice

Therefore the total number of possible outcomes when three dices are thrown

Question 36: A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times ? Five times? times.

Answer:

Total number of possible outcomes when a coin is tossed

If a coin is tossed 3 times, the possible outcomes

If a coin is tossed 4 times, the possible outcomes

If a coin is tossed 5 times, the possible outcomes

If a coin is tossed n times, the possible outcomes

Question 37: How many numbers of four digits can be formed with the digits 1, 2,3,4,5 if the digits can be repeated in the same number?

Answer:

Note: Repetition of the digits is allowed.

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

Maximum number of code combinations

Question 38: How many three digit numbers can be formed by using the digits while each digit may be repeated any number of times?

Answer:

Note: Repetition of the digits is allowed.

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

Maximum number of code combinations

Question 39: How many natural numbers less than can be formed from the digits when a digit may be repeated any number of times?

Answer:

Four digit natural numbers not exceeding can be digits, digits and single digits.

digit natural numbers not exceeding

The first digit can be filled in different ways.

The second digit can be filled in different ways.

The third digit can be filled in different ways.

Therefore the three digit different numbers

digit natural numbers not exceeding

The first digit can be filled in different ways.

The second digit can be filled in different ways.

Therefore the three digit different numbers

Single digit natural numbers not exceeding

The single digit can be filled in different ways. ( is not a natural number)

Therefore the three digit different numbers

Therefore the total number of number

Question 40: How many five digit telephone numbers can be constructed using the digits 0 to 9. If each number starts with 67 and no digit appears more than once?

Answer:

Note: Repetition of the digits is not allowed.

The first digit can be filled in different ways. Only can be chosen.

The second digit can be filled in different ways. Only can be chosen.

The third digit can be filled in different ways.

The fourth digit can be filled in different ways.

The fifth digit can be filled in different ways.

Maximum number of code combinations

Question 41: Find the number of ways in which distinct toys can be distributed among children.

Answer:

Total number of toys

Total number of children

Therefore number of ways in which 8 distinct toys can be distributed among children

Question 42: Find the number of ways in which one can post letters in letter boxes.

Answer:

Total number of letters

Total number of letter boxes

Therefore number of ways in which one can post letters in letter boxes

Question 43: Three dice are rolled. Find the number of possible outcomes in which at least one dice shows .

Answer:

Total number of dice

Therefore the number of possible outcomes

Total number of outcomes when does not show up in any of the dice

Therefore the required number of possible outcomes

Question 44: Find the total number of ways in which balls can be put into boxes so that first box contains just one ball.

Answer:

Total number of balls

Total number of boxes

The first ball can be put in the first box in ways.

Now the remaining balls are to be put in the four boxes. This can be done in ways because there are choices for each ball.

Therefore the required number of ways

Question 45: In how many ways can different balls be distributed among three boxes?

Answer:

Each ball can be distributed in ways.

Therefore the required number of ways different balls be distributed among three boxes

Question 46: In how many ways can letters be posted in letter boxes?

Answer:

Each letter can be posted in different ways.

Therefore the required number of ways to post letters in letter boxes

Question 47: In how many ways can prizes be distributed among students, when

i) no student gets more than one prize ?

ii) a student may get any number of prizes ?

iii) no student gets all the prizes?

Answer:

i) no student gets more than one prize

Number of ways the prizes can be distributed

ii) a student may get any number of prizes

Number of ways the prizes can be distributed

iii) no student gets all the prizes

Number of ways where a student receives all the prices

Number of ways prizes be distributed among students, when no student gets all the prizes

Question 48: There are lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.

Answer:

The hall would be lighted even when one bulb is lighted.

Number of ways the hall can be lighted