FUNDAMENTAL PRINCIPLE OF COUNTING PROBLEMS

Problem 1 :

A mobile phone has a passcode of 6 distinct digits. What is the maximum number of attempts one makes to retrieve the passcode?

Solution :

The passcode must be formed using the following digits

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Each digits must be different.

Total number of ways  =  10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5

  =  151,200

Since we have to use distinct digits, we cannot choose the repeated numbers.

Hence the total number of ways of retrieving the passcode is 151200.

Problem 2 :

Given four flags of different colors, how many different signals can be generated if each signal requires the use of three flags, one below the other?

Solution :

1^st flag may be chosen out of 4 flags, 2^nd flag may be chosen out of 3 flags and 3^rd flag may be chosen out of 2 flags.

Total number of ways  =  4 ⋅ 3 ⋅ 2

  =  24 ways

Hence 24 different signals may be formed using 4 flags.

Problem 3 :

Four children are running a race.

(i) In how many ways can the first two places be filled?

(ii) In how many different ways could they finish the race?

Solution :

(i) Out of 4 children, any one may get first prize. Out of three children, any one may get the second prize.

Hence the total number of ways  =  4 (3)  =  12 ways.

(ii)  Out of 4 ----> Any one may get 1^st prize

Out of 3 ----> Any one may get 2^nd prize

Out of 2 ----> Any one may get 3^rd prize

Remaining 1 will get fourth place.

Hence total number of ways  =  4 ⋅ 3 ⋅ 2 ⋅ 1  =  24 ways

Problem 4 :

Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if (i) repetitions of digits is allowed. (ii) repetitions of digits is not allowed

Solution :

Required three digit number

  ___   ___   ___

(i) repetitions of digits is allowed

Since repetition of digits is allowed, we have 4 options to fill each places.

Hence the numbers to be formed with the given digits are 

  =  4 ⋅ 4 ⋅ 4 

  =   64

(ii) repetitions of digits is not allowed

Hundred place :

We may use any of the digits (2, 4, 6, 8), so we have 4 options.

Tens place :

Repetition is not allowed.So, we have 3 options.

Unit place :

By excluding the number used in the hundreds and tens place, we have 2 options.

Hence total numbers to be formed  =  4 ⋅ 3 ⋅ 2

  =  24

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