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How many five digit numbers can be formed using digits 0, 1, 2, 3, 4

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How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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How many five digit numbers can be formed using digits 0, 1, 2, 3, 4, 5, which are divisible by 3, without any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216

Originally posted by TheRob on 22 Oct 2009, 14:20.
Last edited by Bunuel on 04 Feb 2019, 05:30, edited 1 time in total.
Renamed the topic and edited the question.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 22 Oct 2009, 14:59
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TheRob wrote:
How many five digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, whithout any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216


First step:

We should determine which 5 digits from given 6, would form the 5 digit number divisible by 3.

We have six digits: 0,1,2,3,4,5. Their sum=15.

For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.


Second step:

We have two set of numbers:
{1, 2, 3, 4, 5} and {0, 1, 2, 4, 5}. How many 5 digit numbers can be formed using this two sets:

{1, 2, 3, 4, 5} --> 5! as any combination of these digits would give us 5 digit number divisible by 3. 5!=120.

{0, 1, 2, 4, 5} --> here we can not use 0 as the first digit, otherwise number won't be any more 5 digit and become 4 digit. So, total combinations 5!, minus combinations with 0 as the first digit (combination of 4) 4! --> 5!-4!=96

120+96=216

Answer: E.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 22 Oct 2009, 15:14
Thanks Bunuel for the great explanation . +1 Kudos...

Why don't I borrow your math brain for my GMAT :) LOL
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 24 Oct 2009, 00:06
srini123 wrote:
Thanks Bunuel for the great explanation . +1 Kudos...

Why don't I borrow your math brain for my GMAT :) LOL



I have been thinking the same for quite some days!
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 25 Oct 2009, 14:42
YOu're awesome Bunuel !!
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 26 Oct 2009, 06:21
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Only 2 sets are possible

case (1) 1,2,3,4,5
case (2) 0,1,2,4,5.

case (1) : there will 5! ways to form the number = 120

case (2) ; there will 4*4*3*2*1 = 96 ways

So total no.of ways = 120+96 = 216 ways
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 29 Apr 2010, 23:51
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By the property of divisibility by 3 i.e "a no: is divisible by 3, if the sum of the digits is divisible by 3"(e.g= 12-->1+2=3)

so from 0,1,2,3,4,5 the set of 5 digit no:s that can be formed which is divisible by 3 are 0,1,2,4,5(sum=12) & 1,2,3,4,5(sum=15)

from first set(0,1,2,4,5) no:s formed are 96 i.e first digit can be formed from any 4 no: except 0, second digit from 4 no: except digit used at first place,3rd from rest 3 , 4th from rest 2 no: and in fifth remaining digit since no repetition allowed.

from second set(1,2,3,4,5) no:s formed are 120 i.e first digit can be formed from any 5 digits, second digit from 4 no: except digit used at first place,3rd from rest 3 , 4th from rest 2 no: and in fifth remaining digit since no repetition allowed.

so total 120+96=216
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 06 Feb 2011, 14:48
For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.

i understood the first part but did not get the second part 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3. ..Could you please explain it in a little bit more detail. Thanks
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 06 Feb 2011, 14:55
ajit257 wrote:
For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.

i understood the first part but did not get the second part 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3. ..Could you please explain it in a little bit more detail. Thanks


The sum of the given digits is already a multiple of 3 (15), in order the sum of 5 digits to be a multiple of 3 you must withdraw a digit which is itself a multiple of 3, otherwise (multiple of 3) - (non-multiple of 3) = (non-multiple of 3).
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 06 Feb 2011, 14:59
so lets say we were asked a multiple of 5 so in that case we would have to withdraw the digit 5 ..is that correct ?
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 06 Feb 2011, 15:18
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ajit257 wrote:
so lets say we were asked a multiple of 5 so in that case we would have to withdraw the digit 5 ..is that correct ?


5 or 0, as 0 is also a multiple of 5.

AGAIN: we have (sum of 6 digits)=(multiple of 3). Question what digit should we withdraw so that the sum of the remaining 5 digits remain a multiple of 3? Answer: the digit which is itself a multiple of 3.

Below might help to understand this concept better.

If integers \(a\) and \(b\) are both multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference will also be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=6\) and \(b=9\), both divisible by 3 ---> \(a+b=15\) and \(a-b=-3\), again both divisible by 3.

If out of integers \(a\) and \(b\) one is a multiple of some integer \(k>1\) and another is not, then their sum and difference will NOT be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=6\), divisible by 3 and \(b=5\), not divisible by 3 ---> \(a+b=11\) and \(a-b=1\), neither is divisible by 3.

If integers \(a\) and \(b\) both are NOT multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference may or may not be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=5\) and \(b=4\), neither is divisible by 3 ---> \(a+b=9\), is divisible by 3 and \(a-b=1\), is not divisible by 3;
OR: \(a=6\) and \(b=3\), neither is divisible by 5 ---> \(a+b=9\) and \(a-b=3\), neither is divisible by 5;
OR: \(a=2\) and \(b=2\), neither is divisible by 4 ---> \(a+b=4\) and \(a-b=0\), both are divisible by 4.

Hope it's clear.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 07 Feb 2011, 04:39
3
0,1,2,3,4,5

One digit will have to remain out for all 5 digit numbers;

if 0 is out; Leftover digits will be 1,2,3,4,5 = Sum(1,2,3,4,5)=15.
5! = 120 numbers

if 1 is out; Leftover digits will be 0,2,3,4,5 = Sum(0,2,3,4,5)=14. Ignore(Not divisible by 3)

if 3 is out; Leftover digits will be 0,1,2,4,5 = Sum(0,1,2,4,5)=12.
4*4! = 4*24 = 96

if 4 is out; Leftover digits will be 0,1,2,3,5 = Sum(0,1,2,3,5)=11. Ignore
if 5 is out; Leftover digits will be 0,1,2,3,4 = Sum(0,1,2,3,4)=10. Ignore

Total count of numbers divisible by 3 = 120+96 = 216

Ans: "E"
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post Updated on: 06 Jan 2012, 09:27
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A number is divisible by 3 if sum of its digits is a multiple of 3.

With the given set of digits, there are two possible combinations of 5 digits each-

A. [1,2,3,4,5] No. of possible 5 digit numbers: 5!= 120
B. [0,1,2,4,5] No. of possible 5 digit numbers: 4*4!=96 [the number can't start with a 0]

A+B= 120+96= 216

E

Originally posted by blink005 on 06 Jan 2012, 07:48.
Last edited by blink005 on 06 Jan 2012, 09:27, edited 1 time in total.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 28 Dec 2012, 06:54
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TheRob wrote:
How many five digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, whithout any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216


0 + 1 + 2 + 3 + 4 + 5 = 15

To form 5-digit number, we can remove a digit and the sum should still be divisible by 3.

15 - 1 = 14
15 - 2 = 13
15 - 3 = 12 BINGO!
15 - 4 = 11
15 - 5 = 10

Possible = {5,4,3,2,1} and {5,4,0,2,1}

There are 5! = 120 ways to arrange {5,4,3,2,1}
There are 5! - 5!/5 = 96 ways to arrange {5,4,0,2,1} since 0 cannot start the five number digit.

120 + 96 = 216

Answer: E
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 14 Jan 2013, 00:40
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I did in 1 min 18 sec.
At first I wanted to choose a set of five digits, but started to worry about the complications with the leading zero.

Then I thought that the last digits could always be chosen in only two ways so as to ensure divisibility by three - however, I quickly realized that I would not get all different digits.

Then I realized that once I get a number I can keep permuting the digits while still getting valid numbers.

In an attempt to avoid the leading zero I tried 12345 and noticed that it was divisible by 3. Thus, I've got 5!=120 answers and immediately eliminated two answers, A and B.

Then I addressed the case of a leading zero. Since I wanted to preserve divisibility by 3, I quickly saw that I could only use 0 instead of 3. Thus, the only other possible set was {0, 1, 2, 4, 5}. I tried adding another 5! and got 240, so the answer was slightly less than that.

After that I knew I had to subtract 4!=24 to account for all the possibilities with a leading zero, which left me with 240-24=216. This is how I do such problems...
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 15 Oct 2013, 18:26
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E.

I'm offering up a way to apply the "slot method" to this problem below.

First, as everyone else has identified above, you need to find the cases where the 6 numbers (0, 1, 2, 3, 4, 5) create a 5-digit number divisible by 3.

Shortcut review: a number is divisible by 3 if the sum of the digits in the number is divisible by 3.

So, 12345 would be divisible by 3 (1+2+3+4+5 = 15, which is divisible by 3).

Analyzing the given numbers, we can conclude that only the following two groups of numbers work: 1, 2, 3, 4, 5 (in any order, they would create a five digit number divisible by 3 - confirmed by the shortcut above), and 0, 1,2, 4, 5.

Now we need to count the possible arrangements in both cases, and then add them together.

To use the slot method with case 1 (1,2,3,4,5):
_ _ _ _ _ (five digit number, 5 slots). Fill in the slots with the number of "choices" left over from your pool of numbers. Starting from the left, I have 5 choices I can put in slot #1 (5 numbers from the group 1,2,3,4,5 - pretend I put in number 1, that leaves 4 numbers)
5 _ _ _ _
Fill in the next slot with the number of choices left over (4 choices left...numbers 2 through 5)
5 4 _ _ _
Continue filling out the slots until you arrive at:
5 4 3 2 1
Multiply the choices together: 5x4x3x2x1 (which also happens to be 5!) = 120 different arrangements for the first case.

Now consider case 2 (0,1,2,4,5):
_ _ _ _ _ (five digit number, 5 slots). Here's the tricky part. I can't put 0 as the first digit in the number...that would make it a 4-digit number! So I only have 4 choices to pick from for my first slot!
4 _ _ _ _
Fill in the next slot with the remaining choices (if I put in 1 in the first slot, I have 0,2,4,5 left over...so 4 more choices to go).
4 4 _ _ _
Continue to fill out the slots with the remaining choices:
4 4 3 2 1
Multiply the choices together: 4 x 4! = 96 different arrangements for the second case.

Now the final step is to add all the possible arrangements together from case 1 and case 2:
120 + 96 = 216. And this is our answer.

Hope this alternate "slot" method helps! This is how I try to work these combinatoric problems instead of using formulas... in this case it worked out nicely. Here, order didn't matter (we are only looking for total possible arrangements) in the digits, so we didn't need to divide by the factorial number of slots.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 08 Jun 2014, 09:12
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Bunuel wrote:
TheRob wrote:
How many five digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, whithout any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216


First step:
We should determine which 5 digits from given 6, would form the 5 digit number divisible by 3.

We have six digits: 0,1,2,3,4,5. Their sum=15.

For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.

Second step:

We have two set of numbers:
{1, 2, 3, 4, 5} and {0, 1, 2, 4, 5}. How many 5 digit numbers can be formed using this two sets:

{1, 2, 3, 4, 5} --> 5! as any combination of these digits would give us 5 digit number divisible by 3. 5!=120.

{0, 1, 2, 4, 5} --> here we can not use 0 as the first digit, otherwise number won't be any more 5 digit and become 4 digit. So, total combinations 5!, minus combinations with 0 as the first digit (combination of 4) 4! --> 5!-4!=96

120+96=216

Answer: E.

I tried to do as follows:
take all 5 digit numbers possible : 5 *5*4*3*2
divide by 3 to get all numbers divisible by 3.

What is wrong with this logic?
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 08 Jun 2014, 11:46
ronr34 wrote:
Bunuel wrote:
TheRob wrote:
How many five digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, whithout any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216


First step:
We should determine which 5 digits from given 6, would form the 5 digit number divisible by 3.

We have six digits: 0,1,2,3,4,5. Their sum=15.

For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.

Second step:

We have two set of numbers:
{1, 2, 3, 4, 5} and {0, 1, 2, 4, 5}. How many 5 digit numbers can be formed using this two sets:

{1, 2, 3, 4, 5} --> 5! as any combination of these digits would give us 5 digit number divisible by 3. 5!=120.

{0, 1, 2, 4, 5} --> here we can not use 0 as the first digit, otherwise number won't be any more 5 digit and become 4 digit. So, total combinations 5!, minus combinations with 0 as the first digit (combination of 4) 4! --> 5!-4!=96

120+96=216

Answer: E.

I tried to do as follows:
take all 5 digit numbers possible : 5 *5*4*3*2
divide by 3 to get all numbers divisible by 3.

What is wrong with this logic?


Because the numbers divisible by 3 are not 1/3rd of all possible numbers.

{0, 1, 2, 3, 4} --> 96 5-digit numbers possible with this set.
{0, 1, 2, 3, 5} --> 96 5-digit numbers possible with this set.

{0, 1, 2, 4, 5} --> 96 5-digit numbers possible with this set.
{0, 1, 3, 4, 5} --> 96 5-digit numbers possible with this set.
{0, 2, 3, 4, 5} --> 96 5-digit numbers possible with this set.

{1, 2, 3, 4, 5} --> 120 5-digit numbers possible with this set.

Total = 5*5*4*3*2 = 600 but the numbers which are divisible by 3 come from third and sixth sets: 96 + 120 = 216.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 16 Jun 2014, 23:11
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ronr34 wrote:
I tried to do as follows:
take all 5 digit numbers possible : 5 *5*4*3*2
divide by 3 to get all numbers divisible by 3.

What is wrong with this logic?


We cannot do this because we have the asymmetric 0 as one of the digits. The number of 5 digit numbers that can be formed with 0, 1, 2, 3 and 4 is different from the number of 5 digit numbers that can be formed with 1, 2, 3, 4 and 5 (because 0 cannot be the first digit).

Had the digits been 1, 2, 3, 4, 5 and 6, then your method would have been correct.

If 0 is included:
{0, 1, 2, 3, 4} --> 96 5-digit numbers possible with this set.
{0, 1, 2, 3, 5} --> 96 5-digit numbers possible with this set.
{0, 1, 2, 4, 5} --> 96 5-digit numbers possible with this set. - All these numbers are divisible by 3
{0, 1, 3, 4, 5} --> 96 5-digit numbers possible with this set.
{0, 2, 3, 4, 5} --> 96 5-digit numbers possible with this set.
{1, 2, 3, 4, 5} --> 120 5-digit numbers possible with this set. - All these numbers are divisible by 3
The number of 5 digit numbers in these sets is not the same - Sets with 0 have fewer numbers

If 0 is not included:
{1, 2, 3, 4, 5} --> 120 5-digit numbers possible with this set.
{1, 2, 3, 4, 6} --> 120 5-digit numbers possible with this set.
{1, 2, 3, 5, 6} --> 120 5-digit numbers possible with this set. - All these numbers are divisible by 3
{1, 2, 4, 5, 6} --> 120 5-digit numbers possible with this set.
{1, 3, 4, 5, 6} --> 120 5-digit numbers possible with this set.
{2, 3, 4, 5, 6} --> 120 5-digit numbers possible with this set. - All these numbers are divisible by 3
Here exactly 1/3rd of the numbers will be divisible by 3.
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4  [#permalink]

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New post 17 Sep 2015, 09:42
TheRob wrote:
How many five digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, whithout any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216


Key Knowledge:
(Divisibility Rules, Number Theory) math-number-theory-88376.html
For a number to be divisible by 3, the sum of it's digits must be a multiple of 3


Case Method:

Case#1:
\(\sum 0,1,2,4,5 = 12\)

\(5!=120\) = Number of arrangements of 5 digits
Less:
\(4!=24\) = Number of arrangements of 4 digits
(Removes arrangements where 0 is the leading digit (i.e. fix 0 as leading, arrange the remaining 4 digits 4! ways))

\(5!-4!=96\)

Case #2:
\(\sum 1,2,3,4,5 = 15\)

\(5!=120\) = Number of arrangements of 5 digits

\(120+96=216\)
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Re: How many five digit numbers can be formed using digits 0, 1, 2, 3, 4   [#permalink] 17 Sep 2015, 09:42

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