Question 1 :
To travel from a place A to place B, there are two different bus routes B_{1},B_{2}, two different train routes T_{1}, T_{2} 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.
= 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.
= 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.
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.
The numbers 1, 3, 7 and 9 are not divisible by both 2 and 5.
= 4 numbers
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
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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