GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 26 May 2020, 16:46

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 64144
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

21 Mar 2018, 22:01
00:00

Difficulty:

45% (medium)

Question Stats:

62% (01:31) correct 38% (02:03) wrong based on 123 sessions

### HideShow timer Statistics

How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-digit number so that the two occurrences of the digit 3 are separated by at least one other digit?

(A) 48
(B) 36
(C) 24
(D) 18
(E) 12

_________________
Intern
Joined: 25 Mar 2017
Posts: 1
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

23 Mar 2018, 07:25
2
3
With gaps of one:
1. ( 3 _ 3 _ _ )
2. ( _ 3 _ 3 _ )
3. ( _ _ 3 _ 3 )

With gaps of two:
4. ( 3 _ _ 3 _ )
5. ( _ 3 _ _ 3 )

With gaps of three:
6. (3 _ _ _ 3 )

Total- 6 ways
Remaining numbers: 4, 5, 6- arranged in 3! ways
Therefore 6*3! = 36, Option B
##### General Discussion
Intern
Joined: 14 Mar 2018
Posts: 10
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

21 Mar 2018, 22:09
1
Bunuel wrote:
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-digit number so that the two occurrences of the digit 3 are separated by at least one other digit?

(A) 48
(B) 36
(C) 24
(D) 18
(E) 12

5!/2! - 4! = 120/2 - 24 = 36
B
Intern
Joined: 03 Aug 2016
Posts: 40
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

21 Mar 2018, 22:27
1
1
Option B - 36

Arrangement without restriction = 5!/2!

Arrangement with the restriction where both the I's are together = 4! 2! /2!= 4!

(Method: subtracting what's not allowed from total)

Total = 5!/2! - 4! = 36

Posted from my mobile device

Posted from my mobile device
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2800
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

23 Mar 2018, 11:44
Bunuel wrote:
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-digit number so that the two occurrences of the digit 3 are separated by at least one other digit?

(A) 48
(B) 36
(C) 24
(D) 18
(E) 12

The only way that the two digits are separated by at least one other digit is if they are NOT next to each other. We can use the formula:

(Total number of ways to create the numbers) - (number of ways with the 3’s together) = number of ways to create the numbers with 3’s separated by at least one digit

Using the indistinguishable permutations formula, we note that the two 3’s are indistinguishable. Thus, the total number of ways to create the 5-digit numbers is 5!/2! = 60 ways.

Total number of ways to create the numbers with the 3’s together is 4! = 24 ways.

So, the number of ways to create the numbers with the 3’s separated by at least one digit is 60 - 24 = 36.

_________________

# Jeffrey Miller

Jeff@TargetTestPrep.com
202 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4875
GMAT 1: 770 Q49 V46
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

25 Mar 2018, 15:16
Top Contributor
1
Bunuel wrote:
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-digit number so that the two occurrences of the digit 3 are separated by at least one other digit?

(A) 48
(B) 36
(C) 24
(D) 18
(E) 12

Here's another approach:

Take the task of arranging the 5 digits and break it into stages.

Stage 1: Arrange the 4, 5 and 6
We can arrange n unique objects in n! ways
So, we can arrange these 3 digits in 3! ways (6 ways)
So, we can complete stage 1 in 6 ways

TRICKY PART: We'll now add some spaces where the 3's can be placed.
So, for example, if in stage 1, we arranged three digits as 645, then we'll add spaces before and after each digit.
So, we'd get: _6_4_5_
We will place the two 3's in two of the 4 possible spaces.
This will ENSURE that the 3's are not together.

Stage 2: Select two spaces in which to place the 3's
Since the order in which we select the spaces does not matter, we can use combinations.
We can select 2 spaces from 4 spaces in 4C2 ways (6 ways)
So, we can complete stage 2 in 6 ways

By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus arrange all 5 digits) in (6)(6) ways (= 36 ways)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEOS

_________________
Test confidently with gmatprepnow.com
Intern
Joined: 08 Nov 2019
Posts: 30
GMAT 1: 580 Q47 V22
GPA: 3.29
WE: Architecture (Real Estate)
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

12 May 2020, 18:10
BrentGMATPrepNow wrote:
Bunuel wrote:
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-digit number so that the two occurrences of the digit 3 are separated by at least one other digit?

(A) 48
(B) 36
(C) 24
(D) 18
(E) 12

Here's another approach:

Take the task of arranging the 5 digits and break it into stages.

Stage 1: Arrange the 4, 5 and 6
We can arrange n unique objects in n! ways
So, we can arrange these 3 digits in 3! ways (6 ways)
So, we can complete stage 1 in 6 ways

TRICKY PART: We'll now add some spaces where the 3's can be placed.
So, for example, if in stage 1, we arranged three digits as 645, then we'll add spaces before and after each digit.
So, we'd get: _6_4_5_
We will place the two 3's in two of the 4 possible spaces.
This will ENSURE that the 3's are not together.

Stage 2: Select two spaces in which to place the 3's
Since the order in which we select the spaces does not matter, we can use combinations.
We can select 2 spaces from 4 spaces in 4C2 ways (6 ways)
So, we can complete stage 2 in 6 ways

By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus arrange all 5 digits) in (6)(6) ways (= 36 ways)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEOS

Hi Brent,

Can this be done using restrictions method too ?

5 numbers with 2 3's can be arranged in 5!/2! ways(using MISSISSIPPI rule) = 60
Restriction: allowing 2 3's to be together, the 4 entities can then be arranged in 4! ways = 24

---> number of ways 2 3's will be separated by atleast one other digit : 60-24 = 36

Thoughts ?

Thanks,
K
GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4875
GMAT 1: 770 Q49 V46
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

12 May 2020, 18:19
Top Contributor
mehro023 wrote:
Hi Brent,

Can this be done using restrictions method too ?

5 numbers with 2 3's can be arranged in 5!/2! ways(using MISSISSIPPI rule) = 60
Restriction: allowing 2 3's to be together, the 4 entities can then be arranged in 4! ways = 24

---> number of ways 2 3's will be separated by atleast one other digit : 60-24 = 36

Thoughts ?

Thanks,
K

That's a perfectly valid (and quick!) solution, Karaan.
Nice work!

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
CrackVerbal Quant Expert
Joined: 12 Apr 2019
Posts: 590
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d  [#permalink]

### Show Tags

12 May 2020, 22:50
1
Since the digit 3 is repeated once,
Total number of 5-digit numbers possible = $$\frac{5! }{ 2!}$$ = $$120 / 2$$ = 60.

Of these 60 numbers, there will be numbers where the 3s are together. When some objects are to be considered together, we always take them as one object. If we take the 3s as one object, we have 4 objects in total.
Total number of 5-digit numbers where the 3s are together = 4! = 24.

Therefore, total number of 5-digit numbers where the 3s are separated by at least one digit = 60 – 24 = 36.

The correct answer option is B.

Hope that helps!
_________________
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d   [#permalink] 12 May 2020, 22:50