場合の数

{"meta":{"subjectId":"math","grade":"中学2年","field":"確率","unit":"場合の数","level":"標準","drillId":"math-2g001a"},"cards":[{"id":"1","front":{"title":"順番のある結果を並べる","text":"硬貨を1回目、2回目と区別して投げます。結果の表では表表・表裏・裏表・裏裏の4通りです。\n表裏と裏表は順番が違うので別の結果です。最初の結果を一つ固定し、その後の候補をすべて並べると漏れを防げます。","visual":{"kind":"dataVisual","rows":[["1回目／2回目","表","裏"],["表","表表","表裏"],["裏","裏表","裏裏"]],"alt":"本文の条件を整理した表。"}},"back":{"title":"順番のある結果を並べる","text":"硬貨を2回投げます。1回目が表の結果は{blank:1}通り、裏の結果も{blank:2}通り。全体は{blank:3}通りです。","answers":["2","2","4"],"visual":{"kind":"dataVisual","rows":[["1回目／2回目","表","裏"],["表","表表","表裏"],["裏","裏表","裏裏"]],"alt":"本文の条件を整理した表。"}}},{"id":"2","front":{"title":"戻さないと次の候補が減る","text":"カード1、2、4から1枚ずつ順に取り、戻さず二桁の数を作ります。十の位が1なら12、14の2通りです。\n十の位は3候補、それぞれ一の位は残り2候補なので$3×2=6$通り。12と21は別で、11は同じカードを二度使うので作れません。","visual":{"kind":"dataVisual","rows":[["十の位","できる数"],["1","12・14"],["2","21・24"],["4","41・42"]],"alt":"本文の条件を整理した表。"}},"back":{"title":"戻さないと次の候補が減る","text":"カード2、3、5を戻さず順に2枚取ります。最初は{blank:1}候補、次は{blank:2}候補、二桁の数は{blank:3}通りです。","answers":["3","2","6"],"visual":{"kind":"dataVisual","rows":[["十の位","一の位の候補"],["2","3・5"],["3","2・5"],["5","2・3"]],"alt":"本文の条件を整理した表。"}}},{"id":"3","front":{"title":"戻すと同じものを再び選べる","text":"カード1、2、4を1枚取って戻し、もう一度取ります。最初に1を取っても次の候補は1、2、4です。\n毎回3候補なので$3×3=9$通り。戻す条件なら11も含まれます。戻す・戻さないを先に確認します。","visual":{"kind":"dataVisual","rows":[["1回目／2回目","1","2","4"],["1","11","12","14"],["2","21","22","24"],["4","41","42","44"]],"alt":"本文の条件を整理した表。"}},"back":{"title":"戻すと同じものを再び選べる","text":"カード2、3、5を毎回戻して2回取ります。2回目の候補は{blank:1}個で、全結果は{blank:2}通りです。","answers":["3","9"],"visual":{"kind":"dataVisual","rows":[["1回目／2回目","1","2","4"],["1","11","12","14"],["2","21","22","24"],["4","41","42","44"]],"alt":"本文の条件を整理した表。"}}},{"id":"4","front":{"title":"二つのサイコロを区別する","text":"サイコロA、Bを一回ずつ投げます。Aが1のとき、Bは1から6までの6候補です。\nAの各目について6候補あるため$6×6=36$通り。(A=1,B=2)と(A=2,B=1)は別の結果です。同時に投げてもA、Bを区別して整理できます。","visual":{"kind":"dataVisual","rows":[["A／B","1","2","3","4","5","6"],["1","1,1","1,2","1,3","1,4","1,5","1,6"],["2","2,1","2,2","2,3","2,4","2,5","2,6"],["3","3,1","3,2","3,3","3,4","3,5","3,6"],["4","4,1","4,2","4,3","4,4","4,5","4,6"],["5","5,1","5,2","5,3","5,4","5,5","5,6"],["6","6,1","6,2","6,3","6,4","6,5","6,6"]],"alt":"本文の条件を整理した表。"}},"back":{"title":"二つのサイコロを区別する","text":"サイコロA、Bを投げます。A=3の場合のBは{blank:1}候補、全体の組は{blank:2}通りです。","answers":["6","36"],"visual":{"kind":"dataVisual","rows":[["A／B","1","2","3","4","5","6"],["1","1,1","1,2","1,3","1,4","1,5","1,6"],["2","2,1","2,2","2,3","2,4","2,5","2,6"],["3","3,1","3,2","3,3","3,4","3,5","3,6"],["4","4,1","4,2","4,3","4,4","4,5","4,6"],["5","5,1","5,2","5,3","5,4","5,5","5,6"],["6","6,1","6,2","6,3","6,4","6,5","6,6"]],"alt":"本文の条件を整理した表。"}}}],"questions":[{"id":"1","type":"choice","label":"戻さない場合","question":"1から3までの数字が1枚ずつ書かれたカードを、戻さず順に2枚取り二桁の数を作ります。何通りか選びなさい。","choices":["6通り","9通り","3通り","5通り"],"hint":"二回目は使った1枚を除きます。","explanation":"最初は3候補、次は2候補で3×2=6通り。"},{"id":"2","type":"choice","label":"戻す場合","question":"1から3までの数字のカードを、毎回戻して順に2回取り二桁の数を作ります。何通りか選びなさい。","choices":["9通り","6通り","3通り","5通り"],"hint":"戻すので二回目にも同じ候補があります。","explanation":"3×3=9通りです。"},{"id":"3","type":"choice","label":"戻さない場合","question":"1から4までの数字が1枚ずつ書かれたカードを、戻さず順に2枚取り二桁の数を作ります。何通りか選びなさい。","choices":["12通り","16通り","4通り","7通り"],"hint":"二回目は使った1枚を除きます。","explanation":"最初は4候補、次は3候補で4×3=12通り。"},{"id":"4","type":"choice","label":"戻す場合","question":"1から4までの数字のカードを、毎回戻して順に2回取り二桁の数を作ります。何通りか選びなさい。","choices":["16通り","12通り","4通り","7通り"],"hint":"戻すので二回目にも同じ候補があります。","explanation":"4×4=16通りです。"},{"id":"5","type":"choice","label":"戻さない場合","question":"1から5までの数字が1枚ずつ書かれたカードを、戻さず順に2枚取り二桁の数を作ります。何通りか選びなさい。","choices":["20通り","25通り","5通り","9通り"],"hint":"二回目は使った1枚を除きます。","explanation":"最初は5候補、次は4候補で5×4=20通り。"},{"id":"6","type":"choice","label":"戻す場合","question":"1から5までの数字のカードを、毎回戻して順に2回取り二桁の数を作ります。何通りか選びなさい。","choices":["25通り","20通り","5通り","9通り"],"hint":"戻すので二回目にも同じ候補があります。","explanation":"5×5=25通りです。"},{"id":"7","type":"choice","label":"硬貨二回","question":"硬貨を2回投げたとき、順番を区別した全結果の数を選びなさい。","choices":["4通り","3通り","2通り","6通り"],"hint":"表裏と裏表を区別します。","explanation":"表表・表裏・裏表・裏裏の4通りです。","visual":{"kind":"dataVisual","rows":[["1回目／2回目","表","裏"],["表","表表","表裏"],["裏","裏表","裏裏"]],"alt":"本文の条件を整理した表。"}},{"id":"8","type":"choice","label":"結果の区別","question":"サイコロA、Bで(A=2,B=5)と(A=5,B=2)をどう数えるか選びなさい。","choices":["別々の二つの結果","同じ一つの結果","どちらも数えない","片方だけ数える"],"hint":"A、Bのどちらにどの目が出たかを区別します。","explanation":"AとBの結果が異なるため別の結果です。"},{"id":"9","type":"choice","label":"二つのサイコロ","question":"区別できるサイコロA、Bを一回ずつ投げた結果の組の数を選びなさい。","choices":["36通り","12通り","6通り","21通り"],"hint":"Aの各候補にBの全候補を対応させます。","explanation":"6×6=36通りです。"},{"id":"10","type":"choice","label":"漏れの確認","question":"硬貨2回で「表表・表裏・裏裏」と列挙しました。抜けている結果を選びなさい。","choices":["裏表","表表","表裏","裏裏"],"hint":"一回目が裏のときの二回目を調べます。","explanation":"裏表が抜けています。","visual":{"kind":"dataVisual","rows":[["1回目／2回目","表","裏"],["表","表表","表裏"],["裏","裏表","裏裏"]],"alt":"本文の条件を整理した表。"}},{"id":"11","type":"choice","label":"同じカード","question":"カード1、2、4を戻さず二枚取るとき、作れない数を選びなさい。","choices":["11","12","21","42"],"hint":"同じ一枚を二度使えません。","explanation":"11には1のカードが二枚必要なので作れません。"},{"id":"12","type":"choice","label":"順序の意味","question":"カード1、2、4で12と21を作るとき、数え方を選びなさい。","choices":["異なる二つの数として数える","同じ数として一つにまとめる","どちらも作れない","十の位を無視する"],"hint":"十の位と一の位の役割が違います。","explanation":"数字の順が違うと二桁の数は異なります。"},{"id":"13","type":"choice","label":"二回目の候補","question":"カード1、2、3、4から戻さず1枚取り、最初が2でした。次の候補を選びなさい。","choices":["1、3、4","1、2、3、4","2だけ","1、3だけ"],"hint":"取り出した2を候補から除きます。","explanation":"残りは1、3、4の3枚です。"},{"id":"14","type":"choice","label":"一つを固定","question":"サイコロA、BでA=4と固定したとき、Bの結果は何通りか選びなさい。","choices":["6通り","4通り","12通り","36通り"],"hint":"Bの候補だけを数えます。","explanation":"Bは1から6までの6通りです。"},{"id":"15","type":"order","label":"漏れなく数える","question":"二回の選択の結果を漏れなく数える流れを並べなさい。","items":["順序の区別と戻す条件を確認する","一回目の候補を一つずつ並べる","各候補に対して二回目の全候補を並べる","漏れと重複を点検して全結果を数える"],"hint":"条件により二回目の候補が決まります。","explanation":"条件→一回目→二回目→点検の順です。"}]}