W is the set of all of the four-digit positive integers in which no

Question

W is the set of all of the four-digit positive integers in which no two digits are equal to each other. If a list is made of all the numbers in W arranged in decreasing order, what is the thousands digit of the 1000th integer in the list?

A. 4
B. 5
C. 6
D. 7
E. 8

A. 4
B. 5
C. 6
D. 7
E. 8

Dereno · Answer

Four digit numbers arranged in decreasing order.

Highest four digit number is 9999. But digits repeat, so 9876 is the highest four digit number as per criteria.

Let us take the  thousands place as 9 .

So, we are left with three places, and the digits we have in hand is ( 0,1,2,3,4,5,6,7,8) = 9 digits.

The remaining three places can be filled by 9C3 * 3! = 504.

Lets take the  thousand place as 8 .

So, we are left with three places, and the digits we have in hand is ( 0,1,2,3,4,5,6,7,9) = 9 digits.

The remaining three places can be filled by 9C3 * 3! = 504.

Total numbers till now = 504+504 = 1008.

so, the 1000th digit should be starting with  8.

Option E

vasu1104 · Answer

here numbers are not repitative.
and we want number in decreasing order.
so largest four digit number is 9999. but as we know no num can be same.
so ten thousands digit would be fixed as 9.
then thousand digit would be fixed as 8.

now we need to figure out first how many numb are there between 9801 and 9876.
for that now we need to fix hundreds digit as well and it would be 7.
so for the unit digit we only have 7 choice because 7,8,9 have been used.
so the total num in this range will be 1*1*1*7 = 7

now we will fix hundreds digit as 6. 
 so total numbers in this range =1*1*1*7= 7

now we will fix hundreds digit as 5. 
so the available numbers in this range = 1*1*1*7= 7

up until this we have total num= 7 * 7 *7 =343
we need to find first thousands number and then thousands digit in there.

now lets fix hundreds digit as 5.
so total numbers in this range = 1*1*1*7= 7

so now we have total number 373*7 = 2401
this exceeds our requirements.
but we got that thousands digit of 1000th integer will be 8.

option E

Adit_ · Answer

t misreading the problem again and again when in reality its really straightforward.

They only want you to know the 1000th digit which is EXTREMELY narrow lets see why:

Start first digit as 9:
you have 9*8*7=504 numbers
Start with 8 and another 504 such numbers 
Now you will notice 504*2=1008 and we need the 1000th integer, this means we are yet to cover all the numbers with 1000th digit of 8! 
The last number with 1000th digit of 8 would be 1008th integer, but we are being asked the 1000th, hence its 8

W is the set of all of the four-digit positive integers in which no

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