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PDF Editor FAQ

How many 4 digit numbers greater than 3500 and lower than 5700 can be formed from the digits {2,4, 6,3, 5,8,7}, if repetition of the digits isn't allowed?

Using the J programming language, brute force approach:J primitives: NuVoc - J Wiki#m=./:~n#~(5700>n)*.3500<n=.10#.(4 perm 7){2 3 4 5 6 7 8280The answer is 280.<<<>>>List the first & last few.m3524 3526 3527 3528 3542 3546 3547 3548 3562 3564 3567 3568 3572 3574 3576 3578 3582 3584 3586 3587 3624 3625 3627 3628 3642 3645 3647 3648 3652 3654 ….…. 5467 5468 5472 5473 5476 5478 5482 5483 5486 5487 5623 5624 5627 5628 5632 5634 5637 5638 5642 5643 5647 5648 5672 5673 5674 5678 5682 5683 5684 5687

Let P(x) = (2x^2−2x+1) ^17* (3x^2−3x+1) ^17. What is the sum of all coefficients at odd powers of x?

[math]\left (2x^2−2x+1 \right )^{17} \cdot \left (3x^2−3x+1 \right )^{17}=[/math][math]\left (\left (2x^2−2x+1 \right ) \cdot \left (3x^2−3x+1 \right ) \right )^{17}=[/math][math]\left (\left (2x^2−2x+1 \right ) \cdot 3x^2-\left (2x^2−2x+1 \right ) \cdot 3x+\left (2x^2−2x+1 \right ) \cdot 1 \right )^{17}=[/math][math]\left (\left (6x^4−6x^3+3x^2 \right )-\left (6x^3−6x^2+3x \right )+\left (2x^2−2x+1 \right ) \right )^{17}=[/math][math]\left (6x^4−12x^3+11x^2-5x+1 \right )^{17}[/math]Using Wolfram Alpha and Python:The extended form is:16926659444736 x^68 - 575506421121024 x^67 + 9735650290630656 x^66 - 109202343407714304 x^65 + 913248758871687168 x^64 - 6070953291247779840 x^63 + 33401124436232503296 x^62 - 156364146098740002816 x^61 + 635541896578399764480 x^60 - 2277340727357294051328 x^59 + 7281011235074862956544 x^58 - 20970505521907930939392 x^57 + 54839408752456535900160 x^56 - 131062472157622127394816 x^55 + 287839706400908000501760 x^54 - 583629600225790844559360 x^53 + 1096924191809421985367040 x^52 - 1917651552958240661145600 x^51 + 3127663538560712381022720 x^50 - 4771605090518229975989760 x^49 + 6824925688260322627250688 x^48 - 9170493874774001174956032 x^47 + 11596173041014318510668288 x^46 - 13820886759040335749463552 x^45 + 15546954923719316193043584 x^44 - 16525609248752748698261760 x^43 + 16615734999238247196264768 x^42 - 15816757744245149498259648 x^41 + 14265291485744375337786240 x^40 - 12197930705688322185842304 x^39 + 9893772309541303886482656 x^38 - 7615340305908971006473632 x^37 + 5564233579596878624332134 x^36 - 3860149372683027091729164 x^35 + 2542950184065964590475547 x^34 - 1590807556811390086205173 x^33 + 944960278406017639404001 x^32 - 532910662558628944453032 x^31 + 285250989755046790513012 x^30 - 144869098173165168477956 x^29 + 69774957588162197132864 x^28 - 31853073900563772302568 x^27 + 13773186900239057103806 x^26 - 5636371139362725309026 x^25 + 2180925245433798535218 x^24 - 797058827809329225320 x^23 + 274797687822724777096 x^22 - 89248044492957527536 x^21 + 27261735814675906650 x^20 - 7817888264048734348 x^19 + 2100445377014306777 x^18 - 527473982701111199 x^17 + 123480216786647803 x^16 - 26864082033011368 x^15 + 5412501309383784 x^14 - 1005796653696592 x^13 + 171574661030276 x^12 - 26718545287520 x^11 + 3773320546278 x^10 - 479447191490 x^9 + 54282125026 x^8 - 5410450464 x^7 + 467505916 x^6 - 34317764 x^5 + 2081378 x^4 - 100164 x^3 + 3587 x^2 - 85 x + 1(69 terms)Oks="16926659444736 x^68 - 575506421121024 x^67 + 9735650290630656 x^66 - 109202343407714304 x^65 + 913248758871687168 x^64 - 6070953291247779840 x^63 + 33401124436232503296 x^62 - 156364146098740002816 x^61 + 635541896578399764480 x^60 - 2277340727357294051328 x^59 + 7281011235074862956544 x^58 - 20970505521907930939392 x^57 + 54839408752456535900160 x^56 - 131062472157622127394816 x^55 + 287839706400908000501760 x^54 - 583629600225790844559360 x^53 + 1096924191809421985367040 x^52 - 1917651552958240661145600 x^51 + 3127663538560712381022720 x^50 - 4771605090518229975989760 x^49 + 6824925688260322627250688 x^48 - 9170493874774001174956032 x^47 + 11596173041014318510668288 x^46 - 13820886759040335749463552 x^45 + 15546954923719316193043584 x^44 - 16525609248752748698261760 x^43 + 16615734999238247196264768 x^42 - 15816757744245149498259648 x^41 + 14265291485744375337786240 x^40 - 12197930705688322185842304 x^39 + 9893772309541303886482656 x^38 - 7615340305908971006473632 x^37 + 5564233579596878624332134 x^36 - 3860149372683027091729164 x^35 + 2542950184065964590475547 x^34 - 1590807556811390086205173 x^33 + 944960278406017639404001 x^32 - 532910662558628944453032 x^31 + 285250989755046790513012 x^30 - 144869098173165168477956 x^29 + 69774957588162197132864 x^28 - 31853073900563772302568 x^27 + 13773186900239057103806 x^26 - 5636371139362725309026 x^25 + 2180925245433798535218 x^24 - 797058827809329225320 x^23 + 274797687822724777096 x^22 - 89248044492957527536 x^21 + 27261735814675906650 x^20 - 7817888264048734348 x^19 + 2100445377014306777 x^18 - 527473982701111199 x^17 + 123480216786647803 x^16 - 26864082033011368 x^15 + 5412501309383784 x^14 - 1005796653696592 x^13 + 171574661030276 x^12 - 26718545287520 x^11 + 3773320546278 x^10 - 479447191490 x^9 + 54282125026 x^8 - 5410450464 x^7 + 467505916 x^6 - 34317764 x^5 + 2081378 x^4 - 100164 x^3 + 3587 x^2 - 85 x + 1" sa=s.split(" ") sa=["+"]+sa sa[-3]+="^1" sa+=["x^0"] coef=[int(sa[i-2]+sa[i-1]) for i in range(2, len(sa), 3)][::-1] #so now we have all coefficients in this array print(sum(coef[c] for c in range(1, len(coef), 2))) Output:-88741498560793685913085937 

How do I develop my knowledge in nanotechnology?

Hi, if you are interested in the field of Science like - Biology and Engineering and are enthusiastic about learning about Nano-technology and its applications.Then it would be a good idea to take up a few online courses to enhance your knowledge and get practical exposure through live projects.Nano-technology is a growing field and an in-demand skill, there is a huge scope to excel in this field if you are interested in areas like research and innovation.Here is a list of online courses that are my personal favorites:Nanotechnology - A maker’s course : CourseraThis course will help you if you are looking for practical learning and application.It gives extensive importance to lab demonstrations, practical skills and teaches you the science behind different tools and techniques which can be applied to transform nanotechnology ideas into physical form.Key Features:Flexible deadlines26 hours of course durationCertificate of completion2. Nanoscience and Nano-technology course: VerzeoVerzeo is another great platform if you are looking for nanotechnology courses, it provides you with an opportunity for performing live projects.It is a great course for beginners, the course focuses on detailed insight on fundamental principles of nano-technolgy and the application of it in various fields.Key Features:Duration of the course is 2 monthsConcepts divided into 11 modulesLive projectsCertificate on successful completion of the course.3. Nanotechnology for Health: Innovative Designs for Medical Diagnosis -FutureLearnIf you are interested in medicine and want to learn and explore how the power of nanotechnology can be applied in the healthcare industry to create devices for medical diagnostics, this would be an apt course for you.Key Features:Course duration of 4 weeksLearn at your own paceCertificate on completion4. Learn Engineering Nanomaterials - UdemyThis is a great course on nanotechnology andis available in 2 formats - paid and free.The course focuses on building the foundation by giving us an in-depth understanding on how Nano-technology plays a vital role in material synthesis in a short period.Key Features:Contains 4 modulesAssessment and quizzes52 min of on demand video5. Introduction to modern Nanotechnology: edXThis is a good course for beginners. It focuses on making your basics clear by covering topics ranging from the size and dimensionality of nano systems, nanostructure material characterization methods to electrical and optical characterization techniques.Key Features:Beginner friendlySelf-paced courseYou can add a verified certificate for Rs 3587I hope you like my answer!Image Credits: Verzeo, edX, FutureLearn

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