How many 4 digit even numbers do not use any digit more than

Consider the following cases

Number ending in "0"

Ways to pick number = 1 (fourth digit - is 0) * 9 (third digit - all but 0) * 8 (second digit - all but 0 and third digit) * 7 (first digit - all but 0 and second & third digits)

Number with "0" in tens or hundreds place

Ways to pick number = 4 (fourth digit - {2,4,6,8}) * 2 (either hundreds of tens digit is 0) * 8 (number of ways to pick non-zero of the hundreds/tens digit) * 7 (ways to pick the thousands digit)

Number with no "0"

Ways to pick number = 4 (units digit - no 0 allowed) * 8 (tens digit - one picked and 0 not allowed) * 7 (two picked and 0 not allowed) * 6 (thousands digit - three numbers picked and 0 not allowed)

Total

9*8*7 + 4*2*8*7 + 4*8*7*6 = 56*(9+8+24) = 2296


I suspect this answer is different from the ones you have up there because you are considering numbers starting with 0 as well, which I have not counted

If you allow for 0 in the thousands place, then the answer is easy to find

The number of 4-digit numbers with no repetition is 10*9*8*7=5040

By symmetry half of these will be even and half odd, hence, number of even numbers is 2520

The problem with your solution is that when we have 0 for D then the choices for A, B, and C are 9, 8 and 7 respectively and not 8, 8, and 7. So if we do the way you are doing we would have:

ABCD:
If D is 0, so 1 choice for D, then choices for other letters would be: A-9, B-8, and C-7 --> 1*9*8*7=504;
If D is 2, 4, 6, or 8, so 4 choices for D, then choices for other letters would be: A-8, B-8, and C-7 --> 4*8*8*7=1792;

Total: 504+1792=2296.

Since your solution does not include the numbers that start with 0 (0BCD), you should be getting the same answer as Bunuel and Shrouded.


I have spend about 15 mins on this, was trying to figure out

(1) what were you doing wrong

(2) why can't we simply start from the thousandth digit and move to the right.

I believe the way Bunuel wrote is a little simpler & quicker, as long as you understand that you need to calculate all 4 digit numbers and then subtract the one's that start with zero (or an alternative that I will provide below).

(1) what I think is wrong in your approach (you are getting slightly fewer ways ) -

A B C D ( thousand , hundreds, tens, units)

D can be 0 2 4 6 8 ( any of the 5 digits )

A can be anything except (D or 0) so 8 possibilities - while doing this you have removed extra possibilities. What happens when D itself is 0 ? so you see, some of the A=0 options have already been removed when you said that A can not be same as D.

C can be anything except A and B so 8 possibilities

B can be anything execpt ( A D C ) so 7 possibilities


To compensate for the above, you can do this (this is a correction to your approach or can be an alternative to Bunuel's method) -

D = 4 ways (2,4,6,8 only, 0 has been removed)
A = 8 ways ( can't be equal to D and 0)
B = 8 (can't be equal to A & D)
C = 7 (can't be equal to A, B & D)

Gives = 1792 (4 digits numbers without repetition, but without the ones that end with 0)

D = 1 (can only be equal to 0)
A = 9 (anything but D, which also covers cases where A=0)
B = 8 (anything but A &D)
C = 7 (anything but A, B & D)

Gives = 504 (4 digits numbers without repetition that end with 0)

Total = 1792 + 504 = 2296


Brunel - (2) why can't we simply start from the thousandth digit and move to the right - is this because we need to take the possibilities for the restrictive digits first, such as the ones digit that can only be even ?

Edit - realized after posting that Bunuel had posted a reply.

Yes, D determines the # of choices for the rest of the digits.

It's better to divide D in 2 types (0 and others) i had wrong answer cause I did not separate it then I got 5*9*8*7 for D, C ,B, A.

Tks Bunuel

Somebody please explain me what is wrong woth the below method.

ABCD - 4digit #

Choices for A = 9 (0 is excluded)
Choices for B = 9 (one digit used for A and 0 is now included)
Choices for C = 8
Choices for D = 7

# of 4 digit #s with distinct digits = 9*9*8*7 : note that it would contain equal # of even and odd numbers
hence # of EVEN #s with distinct digits = 9*9*8*7 / 2 = 2268

I think that the red part is not correct.

Consider another example: there are 90 2-digit numbers, out of which 81 have distinct digits (minus 11, 22, 33, 44, 55, 66, 77, 88, 99 total of 9 numbers). Now, if we take your approach then even 2-digit numbers with distinct digits should be half of all 2-digit numbers with distinct digits - 81/2=not an integer.

Actual # is 90/2=45 (even 2-digit numbers) minus 4 (even 2-digt number with same digits: 22, 44, 66, 88) = 41.

Hope it's clear.

thanks for the explanatory notes bunuel.

(10*9*8*7)\2 = 2520

You can't have zero as the first digit of a 4-digit number as in this case it'll become 3-digit number, so you don't have 10 choice for the first digit. Please see the solutions of this problem in the previous posts.

I don't know about others but the very first solution to this problem confused me

Why to start with a zero when it says four digit number , a zero at the beginning will make it a 3 digit no.

here is a more logical approach

4 digit even numbers so first digit cannot be a zero

from 0 to 9 we have 10 digits for each of the four places, please note digits cannot be repeated so each digit has to be distinct.

even numbers will end in a 0 ,2,4,6,8 right?

so lets see if the last digit is zero then

for first digit we have 1 to 9 options{ cannot have 0 here} , second digit we have 8 options , 3 digit we have 7 options and of course units digit i .e has only one option ,0

so 9*8*7*1

same way when last digit is 2

then 8*8*7*1
{ 8 options for the thousands place because it cannot be 2 and 0 }
{ 8 options for the hundredth place because 2 digits have already been reserved }
{ 7 options for the tens place because 3 digits have already been reserved}
{ 1 options for the units place because this case is for when unit digit is 2 }


same way when last digit is 4

then 8*8*7*1 { 8 options for the thousands place because it cannot be 4 and 0 }
{ 8 options for the hundredth place because 2 digits have already been reserved }
{ 7 options for the tens place because 3 digits have already been reserved}
{ 1 options for the units place because this case is for when unit digit is 4 }

when last digit is 6

then 8*8*7*1
{ 8 options for the thousands place because it cannot be 6 and 0 }
{ 8 options for the hundredth place because 2 digits have already been reserved }
{ 7 options for the tens place because 3 digits have already been reserved}
{ 1 options for the units place because this case is for when unit digit is 6 }


when last digit is 8
then then 8*8*7*1 { 8 options for the thousands place because it cannot be 8 and 0 }
{ 8 options for the hundredth place because 2 digits have already been reserved }
{ 7 options for the tens place because 3 digits have already been reserved}
{ 1 options for the units place because this case is for when unit digit is 8 }

so (8*8*7*1)*4 + 9*8*7*1 = 1792 + 504 = 2296

of course explanation is long but in actual exam all I would do is,

9*8*7*1 + (8*8*7*1)4 = 2296

Hope that helps!

I also used the same calculation method...cant figure out my mistake!!!

2296 is a correct answer, so you've made no mistake.

We must caclulate here 5 cases for the last digit: 0,2,4,6 and 8 because only in this case is the 4 digit number even. Important points here: 1st digit cannot be 0, and we are looking for 4 ditinct digits !

Case 1 last digit is equal 0: 9 x 8 x 7 x 1 = 504
Case 2 last digit is equal 2: 8 x 8 x 7 x 1 = 448
Case 3 last digit is equal 4: 8 x 8 x 7 x 1 = 448
Case 4 last digit is equal 6: 8 x 8 x 7 x 1 = 448
Case 5 last digit is equal 8: 8 x 8 x 7 x 1 = 448
--------------------------------------------------------------------------
∑ = 2296

Actually we have a collision of restrictions only between last and first digit for value “0”, so we must test only 2 Cases:
Case 1 last digit is equal 0: 9 x 8 x 7 x 1 = 504
Case 2 last digit-any other even value: 8 x 8 x 7 x 4 = 4x448
∑ = 2296