Method 1: We consider cases, depending on whether the units digit is $0$ or $5$. Case 1: The units digit is $0$. We have to arrange the digits $2, 4, 5, 5$ in the first four positions. Choose two of those four positions for the $5$s, then arrange the remaining two distinct digits in the remaining two positions, which can be done in $$\binom{4}{2}2!$$ ways. Case 2: The units digit is $5$. We have to arrange the digits $0, 2, 4, 5$ in the first four positions. There are three possible positions for the $0$ since it cannot be the leading digit. Once it is placed, arrange the remaining three distinct digits in the remaining three positions. There are $$3 \cdot 3!$$ such numbers. Total: Since the two cases are mutually exclusive and exhaustive, the number of ways the digits of the number $45025$ can be permuted to form a five-digit positive integer that is divisible by $5$ is $$\binom{4}{2}2! + 3 \cdot 3! = 30$$ Method 2: We subtract those integers formed by permuting the digits of $45025$ which are not divisible by $5$ from the $48$ total permutations you found. Those integers have a $2$ or $4$ in the units digit. Choose whether $2$ or $4$ is the units digit. Since $0$ cannot be the leading digit, choose which of the three middle positions is occupied by the $0$. Choose which of the remaining three positions is occupied by whichever of the digits $2$ or $4$ we have not already used. The two $5$s must occupy the remaining two positions. Hence, there are $$2 \cdot 3 \cdot 3 = 18$$ five-digit positive integers that can be formed by permuting the digits of $45025$ which are not divisible by $5$, leaving $$48 - 18 = 30$$ five-digit positive integers that can can be formed by permuting the digits of $45025$ which are divisible by $5$.
Math Expert Joined: 02 Sep 2009 Posts: 86844
How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
00:00
Difficulty:
Question Stats: Hide Show 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 _________________
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
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! waysSo, 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) Answer: BNote: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it. RELATED VIDEOS_________________
Brent Hanneson – Creator of gmatprepnow.comI’ve spent the last 20 years helping students overcome their difficulties with GMAT math, and the biggest thing I’ve learned is… Many students fail to maximize their quant score NOT because they lack the skills to solve certain questions but because they don’t understand what the GMAT is truly testing - Learn more
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] 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 waysRemaining numbers: 4, 5, 6- arranged in 3! ways Therefore 6*3! = 36, Option B
Intern Joined: 14 Mar 2018 Posts: 9
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
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 = 36B
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] Option B - 36Arrangement 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 Joined: 04 Mar 2011 Status:Head GMAT Instructor Affiliations: Target Test Prep Posts: 2814
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
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.Answer: B _________________
Manager Joined: 08 Nov 2019 Posts: 58 Location: United States (MN) Concentration: Finance, International Business Schools: INSEAD Aug 22 intake (A), Stern '24 (WL), Tuck '24 (I) GMAT 1: 580 Q47 V22 GMAT 2: 650 Q49 V29 (Online) GMAT 3: 680 Q49 V33 GMAT 4: 710 Q49 V38 GMAT 5: 730 Q50 V40 GPA: 3.29 WE:Architecture (Other)
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
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! waysSo, 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) Answer: BNote: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it. RELATED VIDEOSHi 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) = 60Restriction: 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 = 36Thoughts ? Thanks,K
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink]
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) = 60Restriction: 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 = 36Thoughts ? Thanks, K That's a perfectly valid (and quick!) solution, Karaan. Nice work!Cheers,Brent _________________
Brent Hanneson – Creator of gmatprepnow.comI’ve spent the last 20 years helping students overcome their difficulties with GMAT math, and the biggest thing I’ve learned is… Many students fail to maximize their quant score NOT because they lack the skills to solve certain questions but because they don’t understand what the GMAT is truly testing - Learn more
CrackVerbal Representative Joined: 03 Oct 2013 Affiliations: CrackVerbal Posts: 4989 Location: India
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink] 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! _________________
Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 11842 GPA: 3.82
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink] Total ways to arrange all given numbers: \(\frac{5! }{ 2!} = \frac{120 }{ 2} = 60\)=> 3's are together: 4, 5, 6, (3,3) = \(\frac{4! * 2! }{ 2!} = \frac{24 }{ 2} = 24\)Number of ways in which 3 are separated: 60 - 24 = 36 Answer B _________________
Re: How many ways can the five digits 3, 3, 4, 5, 6 be arranged into a 5-d [#permalink] 10 Jul 2021, 08:59 |
In how many ways can the digit 3 4579 be arranged among themselves to form a 5-digit even number
Related Posts
Copyright © 2024 themosti Inc.
