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The content applies standard counting techniques such as nCr and nPr formulas, factorial-based counting, and Pascal’s Triangle to count arrangements, paths, committee selections, and constrained cases like vowels kept together or certain routes excluded. Each problem concludes with the computed result and a selected option letter for quick review.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/perms-and-combs-practice-exam-answers/473303/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/perms-and-combs-practice-exam-answers/473303.png","ImageObject",300,407,{"name":92,"@type":93},"Riley","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-10","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What core counting formulas are used throughout the answers?","Question",{"text":112,"@type":113},"The solutions repeatedly use combination and permutation formulas, including nCr and nPr expressed with factorials, and a generating/path formula involving nCk x^{n-k} y^k (tk+1).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the document handle restricted arrangements (e.g., vowels together or avoiding a point)?",{"text":117,"@type":113},"It counts all valid arrangements under the constraint, or subtracts the forbidden cases from the unrestricted total, such as computing totals and then subtracting solutions that pass through a specified point.",{"name":119,"@type":110,"acceptedAnswer":120},"How are Pascal’s Triangle problems solved in this answer set?",{"text":121,"@type":113},"The answers indicate that the number of paths or terms is obtained using Pascal’s Triangle, sometimes with additional instructions like omitting paths through a given point.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},473303,1791359995,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051","Perms puremath30  \nPermsand Combs ‐‐ Answers ‐‐  \nPerms and Combs Practice Exam-ANSWERS  \nThese are the formulas for Perms & Combs you will be given on your diploma  \nn! n Pr = ( n − r ) !  \nn! n Cr = ( n − r ) !r !  \ntk +1 =n Ck xn−ky k  \nANSWERS  \n1. D  \n2. D  \n3. B  \n4. D  \n5. B  \n6. C  \nNR 1. 36  \n7. B  \n8. B  \nNR 2. 10  \n9. A  \n10. A  \n11. A  \n12. C  \nNR 3. 5  \n13. B  \n14. D  \nNR 4. 1440  \n15. A  \n16. C  \n17. B  \n18. C  \n19. B  \n20. A  \nNR 5. 1  \nNR 6. 9375  \n21. C  \n22. B  \n23. A  \n24. A  \nNR 7. 17  \n25. B  \n26. A  \n27. D  \n28. D  \n29. B  \n30. C  \n31. D  \n32. B  \n33. C  \nCopyright © Barry Mabillard, 2006 [www.puremath30.com](www.puremath30.com)  \n1. The number of ways 12 teams can play each other once is 12 C2 . Since each of these combinations happens twice, multiply the result by 2 to find the number of games in total. The answer is 12 C2 × 2 .  \nThe answer is D.  \n2. Draw the letters of KITCHEN, keeping the vowels in a bubble.  \nThere are six items, which can be arranged in 6! ways. The vowels can be arranged in 2! ways inside the bubble. Thus, there are 6!•2! = 1440 ways of arranging the letters keeping the vowels together.  \nThe answer is D.  \n3. There are 2 parents who could go on the left end of the line. Once that parent is placed, only one parent remains for the other end of the line. The remaining six people fill out the middle of the line in any order.  \nMultiplying, the answer is 1440.  \nThe answer is B.  \n4. In the first cube, the possible directions are up, right, and back. Writing as URB, this set of letters can be arranged in 3! = 6 ways. Since the second cube can be written as URB as well, it has 6 possible paths too. To combine the pathways, multiply the results. 6 × 6 = 36 possible paths.  \nThe answer is D.  \n5. There are ten people, and we want six. If Kirsten and James must be on the committee, that reduces the number of available people to 8, and the number of positions remaining on the committee to 4.  \nThe number of possible committees is 8 C4  \nThe answer is B.  \n6. Fill in the pathway using Pascal’s Triangle.  \nThere are 75 paths.  \nThe answer is C.  \nNR 1) Use combinations since we don’t care what order the games are purchased in. 4 C3 ×3 C1 ×2 C2 ×3 C1 = 36  \nThe answer is 36.  \n7. Fill in the pathway using Pascal’s Triangle. Omit the paths passing through point B.  \nThe answer is B.  \n8. The number of lines that can be drawn from four points on a circle is 4 C2 The only option given that has the same solution is the number of ways four people can shake hands once.  \nThe answer is B.  \n(The answer for A is 24 −1 , the answer for C is 6 C2 , and the answer forD is 5)  \nNR 2) In the expansion of ( a + b )n , the number of terms is always one greater than the  \nexponent. So, if ( 3x2 − 2y 3 )3k −9 has 22 terms, that means the exponent must equal 21.  \n3k − 9 = 21  \n3k = 30  \nk = 10  \nThe answer is 10.  \n9. Use combinations since we don’t care what order the committee is.  \n10 C4 ×13 C5  \nThe answer is A.  \n10. Use the formula tk +1 =n Ck xn−kyk to solve this question. First place everything you know into the equation, leaving k blank for now.  \nt+1 =8 C ( mx )8 −  (−4)  \nBy inspection, we get a term containing x4 when k = 4.  \nt4+1 =8 C4 ( mx )8−4 (−4)4  \nt4+1 =8 C4 ( mx )4 (−4)4  \nt = 17920m4 x4  \n5  \nNow plug in the known term in the left side  \n1451520x4 = 17920m4x4  \n1451520 = 17920m4  \n81 = m4  \nm = 3  \nThe answer is A.  \n11. The number of arrangements is 6!•8!3•31!0!•9! = 2.27 × 1017 The answer is A.  \n12. The number of ways the pumpkins and watermelons are together is 4! • 2! = 48 The number of arrangements without restrictions is 5! = 120  \nThe number of ways pumpkins and watermelons  \nare NOT together is 120 – 48 = 72  \nThe answer is C.  \nNR 3) Write n  = 6720 as ~~( ~~n~~ ~~!r~~ ) ~~! = 6720 Write n Cr = 56 as ~~( ~~n~~ ~~−nr!~~) ~~!r! = 56  \nNow divide the expressions to simplify  \nn!  \n(~~  ~~n(~~ ~~n−!r)~~ ~~)!r!~~ ~~! = ~~( ~~n~~ ~~!r~~ ) ~~! × ~~ ~~(~~ ~~n~~ ~~−nr!)~~ ~~!r~~ ~~! = ","cbCaibZaJatA2YLQ","https://ap.wps.com/l/cbCaibZaJatA2YLQ","pdf",312091,13,"English","# Practice Exam - Answers\n## Solution set and final choices","[{\"question\":\"What core counting formulas are used throughout the answers?\",\"answer\":\"The solutions repeatedly use combination and permutation formulas, including nCr and nPr expressed with factorials, and a generating/path formula involving nCk x^{n-k} y^k (tk+1).\"},{\"question\":\"How does the document handle restricted arrangements (e.g., vowels together or avoiding a point)?\",\"answer\":\"It counts all valid arrangements under the constraint, or subtracts the forbidden cases from the unrestricted total, such as computing totals and then subtracting solutions that pass through a specified point.\"},{\"question\":\"How are Pascal’s Triangle problems solved in this answer set?\",\"answer\":\"The answers indicate that the number of paths or terms is obtained using Pascal’s Triangle, sometimes with additional instructions like omitting paths through a given point.\"}]","Perms and Combs Practice Exam - Answers | PDF",1790782231,33]