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How many five-digit numbers can be formed from the digits 0,

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

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New post 29 Aug 2018, 16:25
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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!


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

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New post 23 Sep 2018, 13:23
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!
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 15 Jan 2019, 20:07
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.
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 24 Jan 2019, 17:05
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!
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 24 Jan 2019, 18:59
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?"
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 24 Jan 2019, 19:47
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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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 09 Mar 2019, 14:57
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
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 13 Apr 2019, 19:06
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.

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

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New post 22 Apr 2019, 04:35
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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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 31 Jul 2019, 06:15
04, 12, 20, 24, 32, 40, 52 are the possible numbers

If I take last digit as 0 then possible combinations will be 4x3x2x2x1= 48
If I take last digit as 2 then possible combinations will be 3x3x2x3x1= 54
If I take last digit as 4 and second last digit as 0 then 4x3x2x1x1= 24
If I take last digit as 4 and second last digit as 2 then 3x3x2x1x1= 18

total 144, which is the answer.

However this method took me a lot of time.
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Re: How many five-digit numbers can be formed from the digits 0,  [#permalink]

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New post 10 Sep 2019, 23:42
Bunuel 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


A number to be divisible by 4 its last two digit must be divisible by 4 (similarly a number to be divisible by 2 its last digit must be divisible by 2; to be divisible by 8, last three digits must be divisible by 8 and so on).

Thus the last two digit must be 04, 12, 20, 24, 32, 40, or 52.

If the last two digits are 04, 20, or 40, the first three digits can take 4*3*2= 24 values.
Total for this case: 24*3 = 72.

If the last two digits are 12, 24, 32, or 52, the first three digits can take 3*3*2= 18 values (that's because the first digit in this case cannot be 0, thus we are left only with 3 options for it not 4, as in previous case).
Total for this case: 18*4 = 72.

Grand total 72 +72 =144.
thank you for the explanation.

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

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New post 11 Sep 2019, 05:19
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!


Hey Buddy

Usually for such problems, 0 is only used in the first place if the question states that such usage is allowed and valid.
The question might state is explicitly or convey it in a very subtle manner

If not mentioned, never use 0 in the first place.
Here in this question, we need to make a 5 digit number. If we happen to use 0 in the first place, we can make 01524 as one number. But practically speaking 01524 is equivalent to 1524 and thus is not a 5 digit number

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

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New post 21 Oct 2019, 06:52
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!


Hi,
Well the question asks "How many five-digit numbers can be formed" ,so you can't have a five digit number starting with 0.
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Re: How many five-digit numbers can be formed from the digits 0,   [#permalink] 21 Oct 2019, 06:52

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