{"id":159,"date":"2019-12-27T17:54:18","date_gmt":"2019-12-27T17:54:18","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=159"},"modified":"2020-10-12T17:55:55","modified_gmt":"2020-10-12T17:55:55","slug":"mathcounts-exercises-6","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=159","title":{"rendered":"MATHCOUNTS Exercises &#8211; 6"},"content":{"rendered":"\n<p>How many unique ways to make a change of 63 cents by using any combinations of penny, nickel, dime, and\/or quarter coins? <a onclick='toggle_visibility(\"solution-mathcounts-coins-63-cents\");'>Click here for the solution.<\/a><\/p>\n<div id='solution-mathcounts-coins-63-cents' style='display:none'>\n<p><strong>Solution<\/strong>\nLet&#8217;s assume function \\(f_1(n)\\) is number of ways using pennies to make a change of \\(n\\) cents. Obviously, we have $$f_1(n) =\n\\begin{cases}\n0 &#038; \\text { if n < 0 }\\\\\n1 &#038; \\text { if n = 0 }\\\\\nf_1(n-1) = \\cdots = f_1(0) = 1 &#038; \\text { if n > 0 }\n\\end{cases}\n$$ Lets assume function \\(f_2(n)\\) is number of ways using pennies and\/or nickels to make a change of \\(n\\) cents. $$f_2(n) =\n\\begin{cases}\n0 &#038; \\text { if n < 0 }\\\\\n1 &#038; \\text { if n = 0 }\\\\\nf_1(n)+f_2(n-5) = f_2(n-5) +1 &#038; \\text { if n > 0 }\n\\end{cases}\n$$ Lets assume \\(f_3(n)\\) is number of ways using pennies, nickels and\/or dimes to make a change of \\(n\\) cents. $$f_3(n) =\n\\begin{cases}\n0 &#038; \\text { if n < 0 }\\\\\n1 &#038; \\text { if n = 0 }\\\\\nf_2(n)+f_3(n-10) &#038; \\text { if n > 0 }\n\\end{cases} $$ Lets assume \\(f_4(n)\\) is number of ways using pennies, nickels, dimes and\/or quarters to make a change of \\(n\\) cents. $$f_4(n) =\n\\begin{cases}\n0 &#038; \\text { if n < 0 }\\\\\n1 &#038; \\text { if n = 0 }\\\\\nf_3(n)+f_4(n-25) &#038; \\text { if n > 0 }\n\\end{cases}\n$$ Obviously, the answer for making a changes of 63 cents is the same as to make 60 cents, because 3 pennies must be used to make a change of 3 cents. So we consider \\(n\\) that are multiples of 5, as shown in the following table: $$\n\\begin{array}{c|c|c|c|c}\n\\ n\\  &#038; f_1(n) &#038; f_2(n) &#038; f_3(n) &#038; f_4(n) \\\\\n\\hline\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 1\\\\\n5 &#038; 1 &#038; 2 &#038; 2 &#038; 2\\\\\n10 &#038; 1 &#038; 3 &#038; 4 &#038; 4\\\\\n15 &#038; 1 &#038; 4 &#038; 6 &#038; 6\\\\\n20 &#038; 1 &#038; 5 &#038; 9 &#038; 9\\\\\n25 &#038; 1 &#038; 6 &#038; 12 &#038; 13\\\\\n30 &#038; 1 &#038; 7 &#038; 16 &#038; 18\\\\\n35 &#038; 1 &#038; 8 &#038; 20 &#038; 24\\\\\n40 &#038; 1 &#038; 9 &#038; 25 &#038; 31\\\\\n45 &#038; 1 &#038; 10 &#038; 30 &#038; 39\\\\\n50 &#038; 1 &#038; 11 &#038; 36 &#038; 49\\\\\n55 &#038; 1 &#038; 12 &#038; 42 &#038; 60\\\\\n60 &#038; 1 &#038; 13 &#038; 49 &#038; 73\\\\\n%65 &#038; 1 &#038; 14 &#038; 56 &#038; 87\\\\\n\\end{array}\n$$ Therefore, the answer to the question is: $$ f_4(63) = f_4(60) = 73 $$ \\(\\blacksquare\\)\n<\/p>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How many unique ways to make a change of 63 cents by using any combinations of penny, nickel, dime, and\/or quarter coins? Click here for the solution. Solution Let&#8217;s assume function \\(f_1(n)\\) is number of ways using pennies to make &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=159\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/159"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=159"}],"version-history":[{"count":2,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions"}],"predecessor-version":[{"id":161,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions\/161"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=159"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=159"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}