{"id":4891,"date":"2025-03-05T16:07:00","date_gmt":"2025-03-05T20:07:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4891"},"modified":"2025-03-31T10:11:12","modified_gmt":"2025-03-31T14:11:12","slug":"combination-challenge-2025-02-08","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4891","title":{"rendered":"Combination Challenge – 2025\/03\/05"},"content":{"rendered":"\n

A soccer team has $8$ players. They need to form a starting lineup with one goalkeeper, one captain, one vice-captain, three unique field players, and two bench players. However, two specific players do not want to be captain, vice-captain, or goalie, and another two players do not want to be benched. How many ways can we form the starting lineup? Click here<\/a> for the solution.<\/p>\n\n\n\n

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Solution<\/strong>: Since $2$ players do not want to be benched, they must be included in the starting lineup. Therefore, the other $4$ players in the starting line-up must be selected from the remaining $6$ players, including those $2$ players who do not want to be captain, vice-captain, or goalie.<\/p>\n\n\n\n

Case 1<\/strong>: If none of the two players who do not want to be captain, vice-captain, or goalie is selected.<\/p>\n\n\n\n

The total number of ways to form the starting lineup would be $$6!=720$$ as no restrictions will be applied to the positions.<\/p>\n\n\n\n

Case 2<\/strong>:If we select one of two players who do not want to be captain, vice-captain, or goalie to be in the starting lineup. <\/p>\n\n\n\n

There are $2$ ways to select one of the above two to be one of the $3$ unique field players.<\/p>\n\n\n\n

Additionally, $3$ additional players will be selected from the remaining 4 players who do not have any restrictions. These $3$ players plus the other $2$ players who do not want to be on the bench will have $5!$ permutations for the starting lineup.<\/p>\n\n\n\n

Therefore, the total number of ways is $$ {{2}\\choose{1}}\\cdot {{3}\\choose{1}}\\cdot{{4}\\choose{3}}\\cdot 5!=2\\cdot 3\\cdot 4\\cdot 120=2880$$<\/p>\n\n\n\n

Case 3<\/strong>: If we select both two players who do not want to be captain, vice-captain, or goalie to be in the starting lineup.<\/p>\n\n\n\n

There are $3$ ways to select $2$ positions from the $3$ unique field players for the above two players with $2!$ permutations. <\/p>\n\n\n\n

The remaining $2$ players will be selected from the $4$ players without restriction. These $2$ players plus the $2$ players who do not want to be on the bench will have $4!$ permutations for the starting lineup.<\/p>\n\n\n\n

Therefore the total number of ways is: $${{3}\\choose{2}}\\cdot 2! \\cdot {{4}\\choose{2}}\\cdot 4!=3\\cdot 2\\cdot 6\\cdot 24=864$$<\/p>\n\n\n\n

Combining the above $3$ cases, the answer to the question is $$720+2880+864=\\boxed{4464}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

A soccer team has $8$ players. They need to form a starting lineup with one goalkeeper, one captain, one vice-captain, three unique field players, and two bench players. However, two specific players do not want to be captain, vice-captain, or … 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\/4891"}],"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=4891"}],"version-history":[{"count":22,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4891\/revisions"}],"predecessor-version":[{"id":4914,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4891\/revisions\/4914"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4891"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4891"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4891"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}