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    Add your answer:How many five-digit odd numbers can be formed using the digit 0, 1, 2, 5, 6, 8 if each digit occurs most once in each number?Answer (Detailed Solution Below)GMAT Club Daily Prep Signing up is không lấy phí, quick, and confidential. Join 700,000+ members and get the full benefits of GMAT Club Registration gives you:Applicant StatsBooks/Downloads Hide Show timer Statistics How many odd numbers of 5 digits can be formed with the digits 0 2 3 4 5 without repeating any digit?How many odd numbers of 5 digits can be formed using all the digits from 0 to 9 if repetition of digits allowed?How many 5 digit numbers can be formed from the digits 1,2 3 4 5 using the digits without repetition in how many of them are even?How many 5 digits odd numbers are there?

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∙ 11y ago

Best Answer

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If numbers starting with 0 are permitted, then 6*6*6*6*6 = 7776.

If leading zeos are not allowed, then 5*6*6*6*6 = 6480

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∙ 11y ago

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Q.: How many 5 digit numbers can be formed using the digits 0 2 4 5 8 9 if repetition is allowed?

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How many five-digit odd numbers can be formed using the digit 0, 1, 2, 5, 6, 8 if each digit occurs most once in each number?

192194196198

Answer (Detailed Solution Below)

Option 1 : 192

Concept:

Permutaion
= (fracn!(n – r)!)

Calculation:

⇒ Given digits are 0, 1, 2, 5, 6, 8.

⇒ Consider five blank = _ _ _ _ _

⇒ The number formed is an odd number.

⇒ The unit place can be filled with only 1 or 5 i.e. in 2 ways.

⇒ The number can not begin with ‘0’ Hence first place can be filled in 4 ways.

⇒ The remaining three blanks can be filled with 4 digits in 4P3 ways.

⇒ The total number of
5-digit odd numbers = 2 × 4 × 4P3 = 192

∴ The required result will be 192.

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How many five-digit numbers can
be formed from the digits 0, [#permalink]
  Updated on: 26 Jan 2014, 03:29

00:00

Question Stats:

50%
(02:53) correct 50% (02:45) wrong based on 885 sessions

Hide Show timer Statistics

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

Originally posted by guerrero25 on 25 Jan 2014, 22:54.
Last edited by Bunuel on 26 Jan 2014,
03:29, edited 1 time in total.

Edited the question and added the OA.

Math Expert

Joined: 02 Sep 2009

Posts: 86775

Re: How many five-digit numbers can be formed from the digits 0, [#permalink]
  26 Jan 2014, 04:32

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.

Answer: E.
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How many five-digit numbers can be formed from the digits 0,
[#permalink]
  10 Nov 2022, 06:45

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.

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Joined: 02 Sep 2009

Posts: 86775

Re: How
many five-digit numbers can be formed from the digits 0, [#permalink]
  26 Jan 2014, 04:33

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.

Answer: E.

Similar question to practice:
how-many-five-digit-numbers-can-be-formed-using-digits-91597.html
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Re: How many five-digit numbers can be formed from the digits 0, [#permalink]
  10 May 2014, 06:32

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.

Answer: E.

Dear Bunnel,

i dont understand the highlighted part… when the last 2 digits are 04,
20 or 40 then we have only 3 digits left for the 1st 3 to choose from… as 0,2 or 4 are gone???

same for the next highlighted part.

please explain.

Math Expert

Joined: 02 Sep 2009

Posts: 86775

Re: How many five-digit numbers can be formed from the digits 0,
[#permalink]
  10 May 2014, 06:39

nandinigaur wrote:

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.

Answer: E.

Dear Bunnel,

i dont understand the highlighted part…
when the last 2 digits are 04, 20 or 40 then we have only 3 digits left for the 1st 3 to choose from… as 0,2 or 4 are gone???

same for the next highlighted part.

please explain.

Yes, we are told that no digits can repeat. So, if the last two digits are 04, 20, or 40, we are left with 4 digits for the first, 3 digits for the second and 2 digits for the third one. The same applies to the second case.

Hope it’s clear.
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Re: How many five-digit numbers can be formed from the digits 0,
[#permalink]
  10 May 2014, 11:20

Bunuel wrote:

nandinigaur wrote:

Dear Bunnel,

i dont understand the highlighted part… when the last 2 digits
are 04, 20 or 40 then we have only 3 digits left for the 1st 3 to choose from… as 0,2 or 4 are gone???

same for the next highlighted part.

please explain.

Yes, we are told that no digits can repeat. So, if the last two digits are 04, 20, or 40, we are left with 4 digits for the first, 3 digits for the second and 2 digits for the third one. The same applies to the second case.

Hope it’s clear.

no what i mean is that we have 0, 1, 2, 3, 4, 5 to choose
from. so, if the last two digits are 04, 20, or 40… then 3 of the 6 digits to choose from are gone. we will be left with 2,3,5??? how 4?

Math Expert

Joined: 02 Sep 2009

Posts: 86775

Re: How many five-digit numbers can be formed from the digits 0,
[#permalink]
  11 May 2014, 04:29

nandinigaur wrote:

Bunuel wrote:

nandinigaur wrote:

Dear Bunnel,

i dont understand the highlighted
part… when the last 2 digits are 04, 20 or 40 then we have only 3 digits left for the 1st 3 to choose from… as 0,2 or 4 are gone???

same for the next highlighted part.

please explain.

Yes, we are told that no digits can repeat. So, if the last two digits are 04, 20, or 40, we are left with 4 digits for the first, 3 digits for the second and 2 digits for the third one. The same applies to the second case.

Hope it’s clear.

no what i mean is that we
have 0, 1, 2, 3, 4, 5 to choose from. so, if the last two digits are 04, 20, or 40… then 3 of the 6 digits to choose from are gone. we will be left with 2,3,5??? how 4?

For EACH case of 04, 20, or 40 we used 2 digits and we are left with 4. Isn’t it?
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Re: How many
five-digit numbers can be formed from the digits 0, [#permalink]
  27 May 2014, 16:18

To be divisible by four the number needs to end in 04, 40, 20, 12, 32 or 52.

Now then, we have 4 numbers remaining and 3 slots. 4C3
* 3! ways to order them.
We do this for each combination so: 4C3*3!*6=24*6=144

Answer: E

Hope this helps

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Re: How many 5 digit number combos are divisible by 4? [#permalink]
  06 Oct 2015, 21:40

mitzers 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

Answer is E. Unsure of how to solve this??

You have 6 digits and you need a 5 digit number.

In how many ways can the number be divisible by 4?
If it ends with 04 or 12 or 20 or 24 or 32 or 40 or 52, it will be divisible by 4.

All 3 combinations that have a 0 as one of the last two digits can be formed by using basic counting principle.
4 * 3 * 2= 24. (First digit in 4 ways,
second in3 ways and third in 2 ways)
Since there are 3 such combinations, you get 24*3 = 72

The other 4 combinations which do not have a 0 in the last two digits can be formed by 3 * 3 * 2 = 18
(First digit in 3 ways (no 0), second in 3 ways (leftover 2 digits and 0) and third in 2 ways.
Since there are 4 such combinations, you get 18*4 = 72

Total number of ways = 72 + 72 = 144
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Re: How many five-digit numbers can be formed from the digits 0, [#permalink]   29 Nov 2022, 11:40

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.

Answer: E.

Sir what i dont understand is how can the 3
numbers be arranged in 4 ways in case 2? (highlighted text)

Thanks

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Joined: 02 Sep 2009

Posts: 86775

Re: How many five-digit numbers can be formed from the digits 0, [#permalink]
  29 Nov 2022, 20:01

srishti201996 wrote:

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.

Answer: E.

Sir what i dont understand is how can the 3
numbers be arranged in 4 ways in case 2? (highlighted text)

Thanks

In that case, the last two digit can take 4 different values: 12, 24, 32, or 52. The first three digits can take 18 values. Total = 4*18.
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Re: How many five-digit numbers can be formed from the digits 0, [#permalink]
  13 Apr 2022, 18:06

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]
  06 Jun 2022, 11:30

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.

By far the BEST solution I have seen, given the time limit GMAT has, other solutions are not doable.

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Re: How many five-digit numbers can be formed from the digits 0,
[#permalink]
  30 Jan 2022, 07:01

OE:
The digits given are 0,1,2,3,4, and 5.

The divisibility rule of 4 says, if the last two digits are divisible by 4, the
number is divisible by 4. So, the last two digits can be 04, 12, 20, 24, 32, 40, 52.

If the last two digits are 04 or 20 or 40, then the number of 5-digit numbers possible for each condition is = 4 * 3 * 2 = 24, since all four remaining digits are possible for the first number.

Since this applies for the cases 04, 20, and 40, the total number of 5 digit numbers possible is 3 * 24 = 72 numbers.

Looking the rest of the possibilities, we cannot use 0 as the first digit, so we
will treat these numbers differently. If the last two digits are 12 or 24 or 32 or 52, then the number of 5 digits number possible for each condition is = 3 * 3 * 2 = 18.

Hence the total number of 5 digit number possible = 18*4+72 =144. Thus, the correct answer is E.

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

30 Jan 2022, 07:01

Moderators:

Senior Moderator – Masters Forum

3084 posts

How many odd numbers of 5 digits can be formed with the digits 0 2 3 4 5 without repeating any digit?

45000 no of 5 digit odd number are in this series so , Answer is 45000.

How many odd numbers of 5 digits can be formed using all the digits from 0 to 9 if repetition of digits allowed?

∴ The required result will be 192.

How many 5 digit numbers can be formed from the digits 1,2 3 4 5 using the digits without repetition in how many of them are even?

∴ the total number of arrangements = 120 + 96 + 96 = 312.

How many 5 digits odd numbers are there?

there are 45000 5 digit odd numbers.
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