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CEO  V
Joined: 12 Sep 2015
Posts: 3854
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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Top Contributor
rodri102 wrote:
Hi Bunuel. Could you clarify why we can't use 0 as the first digit? The question doesn't mention this as a restriction.

Thanks!

A 5-digit number can't have 0 in the ten thousands position.
For example, 03212 is not considered a 5-digit number.

If we allowed the first digit to be zero, then even a basic number like 1 could be a 2-digit number (01), a 3-digit number (001), a 4-digit number (0001), etc

Cheers,
Brent
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Intern  B
Joined: 16 Jun 2018
Posts: 15
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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Hi, we need to form 5 digit number , if we start with 0 then we would only be able to form a 4 digit number. (restriction mentioned in question stem). Hope it helps.

rodri102 wrote:
Hi Bunuel. Could you clarify why we can't use 0 as the first digit? The question doesn't mention this as a restriction.

Thanks!
Intern  B
Joined: 28 Aug 2016
Posts: 11
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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guerrero25 wrote:
How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?

A. 36
B. 48
C. 72
D. 96
E. 144

I believe if we just calculate outcome for the first three digits - we will get 5*5*4 or 100 (first digit - we can't use zero, next digit with zero as a possibility, next =4 as the first 2 digits are used). Calculating for any more possibilities should be more than 100, leaving only E as an option.
Senior Manager  S
Joined: 12 Sep 2017
Posts: 298
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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Hello!

Can someone please explain to me why are not we taking the following last two digits?

48, 36, 16, etc...?

Why just must be 04, 12, 20, 24, 32, 40, or 52.

Kind regards!
Intern  B
Joined: 28 Aug 2016
Posts: 11
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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jfranciscocuencag wrote:
Hello!

Can someone please explain to me why are not we taking the following last two digits?

48, 36, 16, etc...?

Why just must be 04, 12, 20, 24, 32, 40, or 52.

Kind regards!

because we don't really have number 6 here:

"How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?"
Intern  B
Joined: 04 Oct 2016
Posts: 17
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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rodri102 wrote:
Hi Bunuel. Could you clarify why we can't use 0 as the first digit? The question doesn't mention this as a restriction.

Thanks!

The question says 5 digit number. Did we ever hear a 5-digit number with a zero in the starting? something like 02367? It becomes a 4 digit number, not 5.
It is not necessary to explicitly mention that 0 cannot be the first digit. That's the ground rule.
Hope you understood!
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Location: Uzbekistan
Schools: Simon '21
GMAT 1: 650 Q47 V33 GPA: 3.5
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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matcarvalho wrote:
guerrero25 wrote:
How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?

A. 36
B. 48
C. 72
D. 96
E. 144

My approach:
Total of 600 possible number: 5x5x4x3x2x1 (zero can't be the 1st digit)
Of those 600 number, 300 are even, because you have 3 odd and 3 even numbers. Of those 300 even numbers, about half are multiple of 4. So answer choice E.

This way is faster than Bunuel's solutions!
Btw you could count even numbers this way: 4*4*3*2*3=288
Target Test Prep Representative D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 6967
Location: United States (CA)
Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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guerrero25 wrote:
How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?

A. 36
B. 48
C. 72
D. 96
E. 144

To be divisible by 4, the last two digits of the number must be divisible by 4. Therefore, they can be 04, 12, 20, 24, 32, 40, and 52. We can split these into two groups: 1) 04, 20, 40, and 2) 12, 24, 32, 52

Group 1:

If the last two digits are 04, then there are 4 choices for the first (or ten-thousands) digit, 3 choices for the second (or thousands) digit, and 2 choices for the third (or hundreds) digit. So we have 4 x 3 x 2 = 24 such numbers if the last two digits are 04. Also there should be 24 numbers if the last two digits are 20 or 40. So we have 24 x 3 = 72 numbers in this group.

Group 2:

If the last two digits are 12, then there are 3 choices for the first (or ten-thousands) digit (since it can’t be 0), 3 choices for the second (or thousands) digit, and 2 choices for the third (or hundreds) digit. So we have 3 x 3 x 2 = 18 such numbers if the last two digits are 12. Also there should be 18 numbers if the last two digits are 24, 32 or 52. So we have 18 x 4 = 72 numbers in this group also.

Therefore, there are a total of 72 + 72 = 144 numbers.

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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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guerrero25 wrote:
How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?

A. 36
B. 48
C. 72
D. 96
E. 144

a good question
for a no to be divisible by 4 , the last 2 digits have to be divisible by 4
possible options 04,12,20,24,32,40,52
we have three options with 0 as units or tens place
so pair possible 4*3*2*1 = 24 and three such pairs are there so 24 * 3 ; 72
also for rest 4 sets ; possible pairs without 0 ; 3*3*2*1 = 18 * 4 ; 72
total pairs; 72 + 72 ; 144
IMO E
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If you liked my solution then please give Kudos. Kudos encourage active discussions. Re: How many five-digit numbers can be formed from the digits 0,   [#permalink] 22 Apr 2019, 04:35

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