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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 How many positive five-digit integers contain the digit tagged by: swerve This topic has 2 expert replies and 0 member replies Top Member How many positive five-digit integers contain the digit Timer 00:00 Your Answer A B C D E Global Stats Difficult How many positive five-digit integers contain the digit grouping "57" (in that order) at least once? For instance, 30,457 and 20,574 are two such integers to include, but 30,475 and 20,754 do not meet the restrictions. A. 279 B. 3,000 C. 3,500 D. 3,700 E. 4,000 The OA is D. Source: Manhattan Prep GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15348 messages Followed by: 1864 members Upvotes: 13060 GMAT Score: 790 Top Reply I believe that the prompt above is outdated; regardless, it does not list a correct answer. The prompt below appears in Manhattan's 5 lb. Book of GRE Practice Problems: Quote: How many times does the digit grouping “57” (in that order) appear in all of the five-digit positive integers? For instance, “57” appears once in 12,357, twice in 57,057, and does not appear in 24,675. A. 279 B. 3,000 C. 3,500 D. 3,700 E. 4,000 Case 1: 57XXX Number of options for the hundreds digit = 10. (Any digit 0-9.) Number of options for the tens digit = 10. (Any digit 0-9.) Number of options for the units digit = 10. (Any digit 0-9.) To combine these options, we multiply: 10*10*10 = 1000. Remaining cases: X57XX, XX57X, XXX57 Number of possible positions for the "57" grouping = 3. (The 3 cases above.) Number of options for the leftmost digit = 9. (Any digit but 0.) Number of options for the rightmost digit = 10. (Any digit 0-9.) Number of options for the last remaining digit = 10. (Any digit 0-9.) To combine these options, we multiply: 3*9*10*10 = 2700. Total ways = Case 1 + Case 2 = 1000 + 2700 = 3700. The correct answer is D. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 swerve wrote: How many positive five-digit integers contain the digit grouping "57" (in that order) at least once? For instance, 30,457 and 20,574 are two such integers to include, but 30,475 and 20,754 do not meet the restrictions. A. 279 B. 3,000 C. 3,500 D. 3,700 E. 4,000 Source: Manhattan Prep The solution posted above is related to another problem (as mentioned). Let´s solve the one originally proposed, in which there are double-counting´s to be dealt with! $$?\,\,:\,\,5{\rm{ - digit}}\,\,{\rm{positive}}\,\,{\rm{integers}}\,\,{\rm{with}}\,\,57{\rm{ - block}}\left( {\rm{s}} \right)$$ \eqalign{ & \left( {\rm{1}} \right)\,\,\,\underline 5 \,\,\, \underline 7 \,\,\, \underline {} \,\,\, \underline {} \,\,\, \underline {} \,\,\,\,\, \to \,\,\,\,{10^3}\,\,{\rm{ways}} \,\, \cr & \left( {\rm{2}} \right)\,\,\,\underline {{\rm{not}}\,0} \,\,\, \underline 5 \,\,\, \underline 7 \,\,\, \underline {} \,\,\, \underline {} \,\,\,\,\, \to \,\,\,\,9 \cdot {10^2}\,\,{\rm{ways}} \,\, \cr & \left. \matrix{ \left( {\rm{3}} \right)\,\,\,\underline {{\rm{not}}\,0} \,\,\, \underline {} \,\,\, \underline 5 \,\,\, \underline 7 \,\,\, \underline {} \,\,\,\,\, \to \,\,\,\,9 \cdot {10^2}\,\,{\rm{ways}} \hfill \cr \left( - \right)\,\,\,\underline {\rm{5}} \,\,\, \underline 7 \,\,\, \underline 5 \,\,\, \underline 7 \,\,\, \underline {} \,\,\,\,\, \to \,\,\,\,10\,\,{\rm{ways}} \hfill \cr} \right\}\,\,\,\, \to \,\,\,\,\,890\,\,{\rm{ways}} \,\, \cr & \left. \matrix{ \left( {\rm{4}} \right)\,\,\,\underline {{\rm{not}}\,0} \,\,\, \underline {} \,\,\, \underline {} \,\,\, \underline 5 \,\,\, \underline 7 \,\,\,\,\, \to \,\,\,\,9 \cdot {10^2}\,\,{\rm{ways}} \hfill \cr \left( - \right)\,\,\,\underline {{\rm{not}}\,0} \,\,\, \underline 5 \,\,\, \underline 7 \,\,\, \underline 5 \,\,\, \underline 7 \,\,\,\,\, \to \,\,\,\,9\,\,{\rm{ways}} \hfill \cr \left( - \right)\,\,\,\underline 5 \,\,\, \underline 7 \,\,\, \underline {} \,\,\, \underline 5 \,\,\, \underline 7 \,\,\,\,\, \to \,\,\,\,10\,\,{\rm{ways}} \hfill \cr} \right\}\,\,\,\, \to \,\,\,\,\,881\,\,{\rm{ways}} \cr} $$? = 1000 + 900 + 890 + 881 = 3671$$ The alternative choices are therefore all wrong. This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for0

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