{"id":4916,"date":"2025-04-21T12:22:23","date_gmt":"2025-04-21T16:22:23","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4916"},"modified":"2025-04-21T13:11:46","modified_gmt":"2025-04-21T17:11:46","slug":"combination-challenge-2025-04-21","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4916","title":{"rendered":"Combination Challenge – 2025\/04\/21"},"content":{"rendered":"\n

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3+18+3$, $18+3+3$, and $3+6+3+9+3$, but not $18+6$ or $24$. (PurpleComet 2023, Middle School, Problem 8<\/em>) Click here<\/a> for the solutions.<\/p>\n\n\n\n

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Solution 1<\/strong>: The problem is equivalent to write $8$ as the sum of at least three positive integers. <\/p>\n\n\n\n

Case 1: There are $8$ $1$s in the sum, i.e. $1+1+1+1+1+1+1+1$. There is only $1$ way.<\/p>\n\n\n\n

Case 2: There are $6$ $1$s in the sum, i.e. $1+1+1+1+1+1+2$. There are $7$ ways.<\/p>\n\n\n\n

Case 3: There are $5$ $1$s in the sum, i.e. $1+1+1+1+1+3$. There are $6$ ways.<\/p>\n\n\n\n

Case 4: There are $4$ $1$ in the sum, with the following sub-cases:<\/p>\n\n\n\n

Case 4.1: For sum as $1+1+1+1+4$, there are $5$ ways.<\/p>\n\n\n\n

Case 4.2: For sum as $1+1+1+1+2+2$, there are $\\dfrac{6!}{4!\\cdot 2!}=15$ ways. <\/p>\n\n\n\n

Case 5: There are $3$ $1$s in the sum, with the following sub-cases:<\/p>\n\n\n\n

Case 5.1: For sum as $1+1+1+5$, there are $4$ ways.<\/p>\n\n\n\n

Case 5.2: For sum as $1+1+1+3+2$, there are $\\dfrac{5!}{3!\\cdot 1!\\cdot 1!}=20$ ways.<\/p>\n\n\n\n

Case 6: There are $2$ $1$s in the sum, with the following sub-cases:<\/p>\n\n\n\n

Case 6.1: For sum as $1+1+6$, there are $3$ ways.<\/p>\n\n\n\n

Case 6.2: For sum as $1+1+4+2$, there are $\\dfrac{4!}{2!\\cdot 1!\\cdot 1}=12$ ways.<\/p>\n\n\n\n

Case 6.3: For sum as $1+1+3+3$, there are $\\dfrac{4!}{2!\\cdot 2!}=6$ ways.<\/p>\n\n\n\n

Case 6.4: For sum as $1+1+2+2+2$, there are $\\dfrac{5!}{2!\\cdot 3!}=10$ ways.<\/p>\n\n\n\n

Case 7: There are $1$ $1$s in the sum, with the following sub-cases:<\/p>\n\n\n\n

Case 7.1: For sum as $1+5+2$, there are $6$ ways.<\/p>\n\n\n\n

Case 7.2: For sum as $1+4+3$, there are $6$ ways.<\/p>\n\n\n\n

Case 7.3: For sum as $1+3+2+2$, there are $\\dfrac{4!}{1!\\cdot 1!\\cdot 2!}=12$ ways.<\/p>\n\n\n\n

Case 8: There is no $1$s in the sum, with the following sub-cases:<\/p>\n\n\n\n

Case 8.1: For sum $2+2+2+2$, there is only $1$ way.<\/p>\n\n\n\n

Case 8.2: For sum $2+2+4$, there are $3$ ways.<\/p>\n\n\n\n

Case 8.3: For sum $2+3+3$, there are $3$ ways.<\/p>\n\n\n\n

Combining all the above cases, the answer to the question is: <\/p>\n\n\n\n

$$1+7+6+5+15+4+20+3+12+6+10+6+6+12+1+3+3=\\boxed{120}$$<\/p>\n\n\n\n

Solution 2<\/strong>: The problem is equivalent to write $8$ as the sum of at least three positive integers, which is the same as placing $8$ identical balls into $n$ unique boxes, where $n=3,4,5,6,7,8$.<\/p>\n\n\n\n

Case 1: Place 8 $1$s into $8$ different boxes, with each box contains at least $1$s. There is only ${{8-1}\\choose{8-1}}=1$ way.<\/p>\n\n\n\n

Case 2: Place 8 $1$s into $7$ different boxes, with each box contains at least $1$s. There are ${{8-1}\\choose{7-1}}=7$ ways.<\/p>\n\n\n\n

Case 3: Place 8 $1$s into $6$ different boxes, with each box contains at least $1$s. There are ${{8-1}\\choose{6-1}}=21$ ways.<\/p>\n\n\n\n

Case 4: Place 8 $1$s into $5$ different boxes, with each box contains at least $1$s. There are ${{8-1}\\choose{5-1}}=35$ ways.<\/p>\n\n\n\n

Case 4: Place 8 $1$s into $4$ different boxes, with each box contains at least $1$s. There are ${{8-1}\\choose{4-1}}=35$ ways.<\/p>\n\n\n\n

Case 5: Place 8 $1$s into $3$ different boxes, with each box contains at least $1$s. There are ${{8-1}\\choose{3-1}}=21$ ways.<\/p>\n\n\n\n

Combining all the above cases, the answer to the question is:<\/p>\n\n\n\n

$$1+7+21+35+35+21=\\boxed{120}$$<\/p>\n\n\n\n

Solution 3<\/strong>: The problem is equivalent to write $8$ as the sum of at least three positive integers. The total ways of writing $8$ as the sum of any positive integers, without any restrictions, including singleton, is $2^{8-1}=128$, as there are at most $8-1=7$ plus signs $(+)$ that can be used in a sum.<\/p>\n\n\n\n

There is $1$ way to write $8$ as the sum of only one positive integers (singleton). And there are $7$ ways to write $8$ as the sum of only two positive integers. <\/p>\n\n\n\n

Therefore, by complementary principle, the answer to the question is: $$128-1-7=\\boxed{120}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3+18+3$, $18+3+3$, and $3+6+3+9+3$, but not $18+6$ or $24$. (PurpleComet 2023, Middle School, Problem 8) Click here … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4916"}],"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=4916"}],"version-history":[{"count":24,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4916\/revisions"}],"predecessor-version":[{"id":4943,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4916\/revisions\/4943"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}