Question

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

Answer 1

let x be such a number
then all number start with 6
hundreds place can be filled be 0,1,2,3,4,5,7,8,9
tens places can be filled by 0,1,2,3,4,5,7,8,9
unit place can be only 0,5 ∵ number is divisible by 5

consider the following cases
1. numbers ending in 0 (6??0)
∵ no digits can be repeated
tens places can be filled by 1,2,3,4,5,7,8,9  
hundreds place can be filled be 1,2,3,4,5,7,8,9
so, 1*8*7*1=56

2. numbers ending in 5 (6??5)
∵ no digits can be repeated
tens places can be filled by 0,1,2,3,4,7,8,9  
hundreds place can be filled be 0,1,2,3,4,7,8,9
so, 1*8*7*1=56
 so the total is 56+56 = 112

Answer 2

Answer:

112

Step-by-step explanation:

We are to find the number of positive integers between 6000 and 7000 which are divisible by 5. So, we are to fill up four vacant places □□□□ by 10 different digits (0 to 9)

satisfying the given conditions and repetition of digits is not allowed.The thousand's place has to be filled up by the digit 6, so there is only one way to fill up this place. As the numbers are divisible by 5, so the unit's place has to be filled up either by 5 or by

0. Thus, there are only 2 ways to fill up unit's place.Now, the hundred's place can be filled up in 8 ways by any of the remaining 8 digits becauseat thousand's place and at unit's place two digits have already been used.16hirmieSimilarly, the ten's place can be filled up in 7 ways.

The number of required positive integers = 1 x8 x 7 x 2 = 112.