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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Picking a 5 digit code with an odd middle digit tagged by: Brent@GMATPrepNow ##### This topic has 3 expert replies and 3 member replies ## Picking a 5 digit code with an odd middle digit Hello, In the following: How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same? A) 420 B) 360 C) 180 D) 120 E) 60 OA: D I was trying to solve as follows: Middle digit could be 1 or 3 or 5. Hence we have, 5 x 4 x 3 x 3 x 2 = 360 Can you please tell me where I am going wrong? Thanks a lot, Sri Senior | Next Rank: 100 Posts Joined 03 Nov 2013 Posted: 34 messages Upvotes: 7 Sri, There are six numbers 1, 2, 3, 4, 5, 6 and five digits code. So let us first pick the digit with restrictions ie the middle digit. Since it has to be an odd number it can be only be filled with 1, 3, 5 (3 Choices). For other four digits no number can repeat so. Next digit, let us say we start with 1st, it would have 6-1 = 5 choices 6 (total numbers) - 1(number used as middle digit) Similarly the 2nd would have 4 choices. 3rd is the middle digit, it has 3 choices. 4th would have 2 choices. 5th one. So the answer would be 5 X 4 X 3 X 2 X 1= 120. ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 gmattesttaker2 wrote: How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same? A) 420 B) 360 C) 180 D) 120 E) 60 OA: D I was trying to solve as follows: Middle digit could be 1 or 3 or 5. Hence we have, 5 x 4 x 3 x 3 x 2 = 360 Can you please tell me where I am going wrong? Thanks a lot, Sri Hi Sri, Your solution looks good. For more clarity, let's examine it step by step. Take the task of building 5-digit numbers and break it into stages. We'll start with the most restrictive stage. Stage 1: Select the middle digit This digit can be 1, 3 or 5, so we can complete stage 1 in 3 ways Stage 2: Select the 1st digit There are now 5 digits remaining to choose from, so we can complete this stage in 5 ways Stage 3: Select the 2nd digit There are now 4 digits remaining to choose from, so we can complete this stage in 4 ways Stage 4: Select the 4th digit There are now 3 digits remaining to choose from, so we can complete this stage in 3 ways Stage 5: Select the 5th digit There are now 2 digits remaining to choose from, so we can complete this stage in 2 ways By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus build a 5-digit number) in (3)(5)(4)(3)(2) ways (= 360 ways) Cheers, Brent Aside: For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting?id=775 _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 gmattesttaker2 wrote: How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same? A) 420 B) 360 C) 180 D) 120 E) 60 Here's another approach. IGNORE the restriction about the middle digit being odd. So, we'll create 5-digit numbers where no digits are repeated. Take the task of building 5-digit numbers and break it into stages. Stage 1: Select the 1st digit There are 6 digits to choose from, so we can complete this stage in 6 ways Stage 2: Select the 2nd digit There are now 5 digits remaining to choose from, so we can complete this stage in 5 ways Stage 3: Select the 3rd digit There are now 4 digits remaining to choose from, so we can complete this stage in 4 ways Stage 4: Select the 4th digit There are now 3 digits remaining to choose from, so we can complete this stage in 3 ways Stage 5: Select the 5th digit There are now 2 digits remaining to choose from, so we can complete this stage in 2 ways By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus build a 5-digit number) in (6)(5)(4)(3)(2) ways = 720 ways IMPORTANT: Since the digits are equally distributed and since half of the 6 digits are odd, we can conclude that HALF of the 720 5-digit numbers have an ODD middle digit, and HALF have an EVEN middle digit, So, the number of 5-digit numbers with an ODD middle digit = 720/2 = 360 = B Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 Hi Dblooos, There's a small problem with your solution ... Dblooos wrote: Sri, There are six numbers 1, 2, 3, 4, 5, 6 and five digits code. So let us first pick the digit with restrictions ie the middle digit. Since it has to be an odd number it can be only be filled with 1, 3, 5 (3 Choices). At this point, you have selected the middle digit For other four digits no number can repeat so. Next digit, let us say we start with 1st, it would have 6-1 = 5 choices 6 (total numbers) - 1(number used as middle digit) Similarly the 2nd would have 4 choices. 3rd is the middle digit, it has 3 choices. you have already selected the middle digit, so you don't need this step 4th would have 2 choices. 5th one. So the answer would be 5 X 4 X 3 X 2 X 1= 120. Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Senior | Next Rank: 100 Posts Joined 03 Nov 2013 Posted: 34 messages Upvotes: 7 Yes, Brent you are right. Sri, I am sorry I was wrong. As Brent highlighted it should be 360. Legendary Member Joined 14 Feb 2012 Posted: 641 messages Followed by: 8 members Upvotes: 11 Brent@GMATPrepNow wrote: gmattesttaker2 wrote: How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same? A) 420 B) 360 C) 180 D) 120 E) 60 OA: D I was trying to solve as follows: Middle digit could be 1 or 3 or 5. Hence we have, 5 x 4 x 3 x 3 x 2 = 360 Can you please tell me where I am going wrong? Thanks a lot, Sri Hi Sri, Your solution looks good. For more clarity, let's examine it step by step. Take the task of building 5-digit numbers and break it into stages. We'll start with the most restrictive stage. Stage 1: Select the middle digit This digit can be 1, 3 or 5, so we can complete stage 1 in 3 ways Stage 2: Select the 1st digit There are now 5 digits remaining to choose from, so we can complete this stage in 5 ways Stage 3: Select the 2nd digit There are now 4 digits remaining to choose from, so we can complete this stage in 4 ways Stage 4: Select the 4th digit There are now 3 digits remaining to choose from, so we can complete this stage in 3 ways Stage 5: Select the 5th digit There are now 2 digits remaining to choose from, so we can complete this stage in 2 ways By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus build a 5-digit number) in (3)(5)(4)(3)(2) ways (= 360 ways) Cheers, Brent Aside: For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting?id=775 Hello Brent, Thanks a lot for your excellent explanation (as always). 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