# HOW MANY NUMBERS ARE THERE BETWEEN 1 AND 1000

How many numbers are there between 1 and 1000 :

Here we are going to see some practice questions based on  the concept fundamental principle of counting.

## How many numbers are there between 1 and 1000 - Practice questions

Question 1 :

To travel from a place A to place B, there are two different bus routes B1,B2, two different train routes T1, T2 and one air route A1. From place B to place C there is one bus route say B'1, two different train routes say T'1, T'2 and one air route A'1. Find the number of routes of commuting from place A to place C via place B without using similar mode of transportation.

Solution :

CASE 1 :

If we choose the transportation bus from place A to B, we have to choose either train or air, to go from B to C.

• If we choose B1, we will have 3 ways (T1', T2', A1')
• If we choose B2, we will have 3 ways (T1', T2',A1')

=  3 + 3  =  6  ways

CASE 2 :

If we choose the transportation train from place A to B, we have to choose either bus or air, to go from B to C.

• If we choose T1, we will have 2 ways (B1', A1')
• If we choose T2, we will have 2 ways (B1', A1')

=  2 + 2   =  4 ways

CASE 3 :

If we choose the transportation air from place A to B, we have to choose either bus or train, to go from B to C.

• If we choose A1, we will have 3 ways (B1', T1', T2')

Hence total number of routes  =  6 + 4 + 3

=  13 ways

Question 2 :

How many numbers are there between 1 and 1000 (both inclusive) which are divisible neither by 2 nor by 5?

Solution :

We may form one digit, two digit and three digit numbers from 1 to 1000.

## One digit numbers not divisible by 2 and 5

The numbers 1, 3, 7 and 9 are not divisible by both 2 and 5.

=  4 numbers

## Two digit numbers not divisible by 2 and 5

Since the required numbers are not divisible by bot 2 and 5, it ends with (1, 3, 7, 9)

Unit digit :

We have 4 options

Tens digit :

Other than 0, we have 9 options

=  9  ⋅ 4  =  36 numbers

## Three digit numbers not divisible by 2 and 5

Since the required numbers are not divisible by bot 2 and 5, it ends with (1, 3, 7, 9)

Unit digit :

We have 4 options

Hundreds digit :

Other than 0, we have 9 options

Tens digit :

Including 0, we have 10 options

=  10 ⋅ 9 ⋅ 4  =  360 numbers

Hence total number to be formed  =  4 + 36 + 360

=  400 numbers

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