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Video, faktöriyel hesaplamaları, \"rakamları farklı\" kavramı, kitapların sıralanması, aile üyeleri oturma düzeni ve spor toto gibi günlük hayattan örneklerle ilgili olasılık problemlerini içermektedir."]},"endTime":6792,"title":"Permütasyon, Kombinasyon ve Olasılık Matematik Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Permütasyon ve Kombinasyon Arasındaki Fark","list":{"type":"unordered","items":["Kombinasyon sadece seçim yapmak demektir, kimseyi sıralamazsınız.","Permütasyonda hem seçim yapar hem de sıralama yapılır.","Permütasyon \"seç ve sırala\" anlamına gelirken, kombinasyon sadece ve sadece seçim yapmak demektir."]},"beginTime":0,"endTime":67,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=0&ask_summarization=1"},{"index":1,"title":"Permütasyon ve Kombinasyon Formülleri","list":{"type":"unordered","items":["Sıralama sorularında faktöriyel kullanılır (örneğin 5 kişi düz bir sırada kaç farklı şekilde oturur? 5 faktöriyel).","Seçim sorularında kombinasyon formülü kullanılır (örneğin 10 kişiden 3 kişi kaç farklı şekilde seçilir? 10'ın 3'lüsü).","Kombinasyon formülünde, n'ın r'li kombinasyonu hesaplanırken n'den r adım geriye gidilir ve alttaki de r kadar geriye gidilir."]},"beginTime":67,"endTime":160,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=67&ask_summarization=1"},{"index":2,"title":"Kombinasyon Formülünün Uygulanması","list":{"type":"unordered","items":["Kombinasyon formülünde büyük sayı yerine küçük sayı kullanmak daha kolaydır (örneğin 20'ın 18'isi yerine 20'ın 2'si kullanılır).","n'nin 0'ı 1'e, n'nin 1'i n'e eşittir.","Permütasyonda hem seçim hem sıralama yapılırken, kombinasyonda sadece seçim yapılır."]},"beginTime":160,"endTime":278,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=160&ask_summarization=1"},{"index":3,"title":"Permütasyon ve Kombinasyon Örnekleri","list":{"type":"unordered","items":["Permütasyon sorularında \"bulunmaz\" durumunda, belirtilen elemanlar çıkarılıp kalan elemanlar arasından seçim ve sıralama yapılır.","Kombinasyon sorularında \"bulunmaz\" durumunda, belirtilen elemanlar çıkarılıp kalan elemanlar arasından sadece seçim yapılır.","\"Veya\" durumunda, tüm durumdan ikisinin olmadığı aksi durum çıkarılarak çözüm bulunur."]},"beginTime":278,"endTime":601,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=278&ask_summarization=1"},{"index":4,"title":"Permütasyon ve Kombinasyon Arasındaki Fark","list":{"type":"unordered","items":["Permütasyonda hem seçim hem sıralama yapılırken, kombinasyonda sadece seçim yapılır.","Permütasyonda faktöriyel kullanılırken, kombinasyonda faktöriyel bölü faktöriyel şeklinde hesaplama yapılır.","Permütasyon formülü P'nin üste yüklenmesiyle gösterilirken, kombinasyon formülü C'nin hem üste hem alta yüklenmesiyle gösterilir."]},"beginTime":602,"endTime":835,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=602&ask_summarization=1"},{"index":5,"title":"Ana Formüller","list":{"type":"unordered","items":["Permütasyon, kombinasyon ve olasılık problemlerinde \"ve\" kelimesi çarpma anlamına gelirken, \"veya\" kelimesi toplama anlamına gelir.","Soru içerisinde verilen görev bitene kadar daima çarpma yapılır, yeni bir alternatif yol varsa toplama devreye girer.","Görev bitene kadar çarpma yapılır, görev bittiğinde ve alternatif yol varsa toplama yapılır."]},"beginTime":835,"endTime":1280,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=835&ask_summarization=1"},{"index":6,"title":"Yolculuk Problemi","list":{"type":"unordered","items":["A'dan C'ye gidip gelme problemi, farklı yollarla gidip gelme olasılıklarını hesaplamayı içerir.","Kişi kısayoldan gidip kısayoldan gelebilir, kısayoldan gidip uzundan gelebilir, uzundan gidip uzundan gelebilir veya uzundan gidip kısayoldan gelebilir.","Her bir yol seçeneği için çarpma işlemi yapılır, görev bittiğinde toplama işlemi devreye girer."]},"beginTime":1286,"endTime":1415,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=1286&ask_summarization=1"},{"index":7,"title":"Mektup ve Posta Kutusu Problemleri","list":{"type":"unordered","items":["Dört farklı mektubu beş posta kutusuna postalamak için, görev bitene kadar daima çarpma işlemi yapılır.","Her mektubu farklı posta kutusuna atma veya bir posta kutusuna en çok bir mektup atma durumunda, mektup sayısı kadar azalan sayılarla çarpma yapılır.","Her kutuya en az bir mektup atma problemi, farklı dağılım durumlarını (2-2-1, 3-1-1) hesaplayarak çözülür."]},"beginTime":1415,"endTime":1726,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=1415&ask_summarization=1"},{"index":8,"title":"Permütasyon Kullanımı","list":{"type":"unordered","items":["Zor soru sınıfına giren problemde, üç posta kutusuna her kutuda en az bir mektup olacak şekilde dağılım durumları hesaplanır.","Tekrarlı permütasyon kullanılarak, farklı dağılım durumları (2-2-1, 3-1-1) için faktöriyel hesaplamaları yapılır.","A sınıfında 10 öğrenci, B sınıfında 15 öğrenci, C sınıfında 12 öğrenci olduğu bilgisi verilerek yeni bir problem başlangıcı yapılır."]},"beginTime":1726,"endTime":1770,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=1726&ask_summarization=1"},{"index":9,"title":"Okul Başkan ve Yardımcı Seçimi","list":{"type":"unordered","items":["Okul başkanı ve yardımcısı seçimi için 37 öğrenci arasından başkan seçildikten sonra 36 öğrenci arasından yardımcı seçilebilir, toplam 37×36 farklı seçim olasılığı vardır.","Aynı sınıftan başkan ve yardımcının seçilmesi durumunda, A sınıfından 10×9, B sınıfından 15×14 ve C sınıfından 12×11 farklı seçim olasılığı vardır.","Permütasyon ve kombinasyon konuları birbirinden ayrılmaz bütünlerdir ve bu sorular ortak olarak ele alınabilir."]},"beginTime":1777,"endTime":2008,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=1777&ask_summarization=1"},{"index":10,"title":"Cevap Anahtarı Soruları","list":{"type":"unordered","items":["20 sorudan oluşan, her biri 5 şıklı bir sınavın cevap anahtarı 5²⁰ farklı şekilde hazırlanabilir.","Ard arda iki şık aynı olmamak koşuluyla cevap anahtarı 5×4¹⁹ farklı şekilde hazırlanabilir.","Ard arda üç şık aynı olmamak koşuluyla cevap anahtarı 5×4×3¹⁷×5×4×3¹⁶×...×5×4×3¹⁸=5×4×3¹⁸ farklı şekilde hazırlanabilir."]},"beginTime":2008,"endTime":2326,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2008&ask_summarization=1"},{"index":11,"title":"Üç Basamaklı Sayılar Soruları","list":{"type":"unordered","items":["Soruda bir, iki, üç, dört, beş, altı rakamları kullanılarak üç basamaklı sayılar oluşturuluyor ve rakamları farklı olma şartı verilmemiş.","Üç basamaklı tek sayılar için birler basamağı 1, 3 veya 5 olabilir, yüzler basamağı 1'den başlayarak 6'ya kadar, onlar basamağı ise 7 farklı rakamdan biri gelebilir.","Çift sayılar için birler basamağı 0, 2, 4 veya 6 olabilir, ancak 246 dışında başka çift sayı oluşturulamaz çünkü rakamları farklı olma şartı verilmemiş."]},"beginTime":2328,"endTime":2438,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2328&ask_summarization=1"},{"index":12,"title":"Rakamları Farklı Olma Şartı","list":{"type":"unordered","items":["Rakamları farklı olma şartı verilmediği sürece sıfırın ayrı düşünülmesi gerekmez, ancak rakamları farklı olma şartı verildiğinde sıfır ayrı düşünülmelidir.","Üç basamaklı rakamları farklı sayılar için sıfırın yüzler basamağına gelmesi imkansızdır, bu nedenle sıfır ayrı düşünülmelidir.","Üç basamaklı rakamları farklı sayılar için 4 farklı rakam kullanılabilir, yüzler basamağı 4, 5, 6 olabilir, onlar basamağı 7 farklı rakamdan biri gelebilir."]},"beginTime":2438,"endTime":2449,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2438&ask_summarization=1"},{"index":13,"title":"Özel Şartlı Sayılar","list":{"type":"unordered","items":["Beş ile bölünebilen sayılar için birler basamağı 0 veya 5 olmalıdır, yüzler basamağı 1'den başlayarak 6'ya kadar, onlar basamağı ise 7 farklı rakamdan biri gelebilir.","Üçyüz'den büyük sayılar için yüzler basamağı 3, 4, 5 veya 6 olabilir, onlar basamağı 7 farklı rakamdan biri gelebilir, birler basamağı da 7 farklı rakamdan biri gelebilir.","Üçyüz sayısının kendisi de dahil olduğu için toplamda bir sayı çıkarılmalıdır."]},"beginTime":2449,"endTime":2535,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2449&ask_summarization=1"},{"index":14,"title":"Özel Şartlı Sayılar (Devam)","list":{"type":"unordered","items":["Üçyüz'den büyük ve çift sayılar için yüzler basamağı 3, 4, 5 veya 6 olabilir, onlar basamağı 7 farklı rakamdan biri gelebilir, birler basamağı ise 0, 2, 4 veya 6 olabilir.","Üçyüz'den büyük ve tek sayılar için yüzler basamağı 3, 4, 5 veya 6 olabilir, onlar basamağı 7 farklı rakamdan biri gelebilir, birler basamağı ise 1, 3 veya 5 olabilir.","Tek sayılar için üçyüz sayısının kendisi dahil olmadığı için çıkarılma işlemi gerekmez."]},"beginTime":2535,"endTime":2620,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2535&ask_summarization=1"},{"index":15,"title":"Özel Aralıklar","list":{"type":"unordered","items":["Üçyüzkırk'tan büyük sayılar için önce yüzün katları ayrı düşünülür, sonra diğer sayılar ayrı değerlendirilir.","Üçyüzkırk ile üçyüz arasında sadece 2 gelir, üçyüz'den büyükler için 3, 4, 5 veya 6 gelir.","Sayıların üçyüzkırk olma ihtimali olduğundan, toplamda bir sayı çıkarılmalıdır."]},"beginTime":2620,"endTime":2752,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2620&ask_summarization=1"},{"index":16,"title":"Rakamları Farklı Kavramı","list":{"type":"unordered","items":["\"Rakamları farklı\" dediğinde her rakamı ayrı düşünmek zorundasınız, kullanılmış bir rakamı tekrar kullanamazsınız.","Sıfır varsa, sıfırı ayrı düşünmek gerekir çünkü sıfır başa gelemez.","Rakamları farklı olduğunda, her basamağı için kalan rakamların sayısını hesaplayarak çözüm yapılır."]},"beginTime":2753,"endTime":2808,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2753&ask_summarization=1"},{"index":17,"title":"Üç Basamaklı Sayılar Oluşturma","list":{"type":"unordered","items":["Üç basamaklı sayı oluştururken, başa sıfır gelemez, bu nedenle ilk basamak için 6 seçenek, diğer basamaklar için kalan rakamlar kullanılır.","Tek sayılar oluşturmak için birler basamağı için 3 seçenek (1, 3, 5) vardır ve diğer basamaklar için kalan rakamlar kullanılır.","Çift sayılar oluşturmak için birler basamağı için 3 seçenek (0, 2, 4, 6) vardır ve diğer basamaklar için kalan rakamlar kullanılır."]},"beginTime":2808,"endTime":3026,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=2808&ask_summarization=1"},{"index":18,"title":"Özel Durumlar ve Çözüm Örnekleri","list":{"type":"unordered","items":["Beş'e bölünen sayılar için birler basamağı 0 veya 5 olmalıdır, bu durumda sıfır ayrı düşünülmelidir.","Üçyüz'den büyük sayılar için yüzler basamağı 3, 4, 5 veya 6 olabilir ve diğer basamaklar için kalan rakamlar kullanılır.","İkiyüzkırk'tan küçük sayılar için, ikiyüzkırk ile ikiyüz arasında olan sayılar ve ikiyüz'den küçük olan sayılar ayrı ayrı düşünülmelidir."]},"beginTime":3026,"endTime":3297,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3026&ask_summarization=1"},{"index":19,"title":"Permütasyon Problemi Çözümü","list":{"type":"unordered","items":["Soruda 300'den büyük ve 3, 4, 5, 6 rakamlarından oluşan çift sayılar bulunması isteniyor.","Çift sayılar için son rakam 2, 4 veya 6 olabilir, ancak 4 ve 6 ortak rakamlar olduğu için ayrı hesaplanmalı.","Çözümde önce 4 ve 6 rakamları hariç diğer rakamlarla permütasyon hesaplanıyor, sonra 4 ve 6 için ayrı hesaplama yapılıyor."]},"beginTime":3300,"endTime":3469,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3300&ask_summarization=1"},{"index":20,"title":"Permütasyon Sorularının Mantığı","list":{"type":"unordered","items":["Öğretmen, permütasyon sorularının ezberlenmemesi gerektiğini, mantığının anlaşılmaması gerektiğini vurguluyor.","Farklı (özdeş olmayan) 3 matematik, 4 kimya ve 5 fizik kitabı bir rafa kaç farklı şekilde dizilebileceği sorusu veriliyor.","Tüm kitapların bir arada dizilmesi durumunda cevap 12 faktöriyel olarak hesaplanıyor."]},"beginTime":3469,"endTime":3575,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3469&ask_summarization=1"},{"index":21,"title":"Farklı Permütasyon Soruları","list":{"type":"unordered","items":["Fizik kitaplarının birbirine bağlanarak tek bir kitap gibi davranması durumunda, 5 fizik kitabı birleştirilerek 1 sayı gibi görülüyor.","Geriye kalan 3 matematik ve 4 kimya kitabıyla birlikte toplam 8 kitap olduğu için cevap 8 faktöriyel çarpı 5 faktöriyel olarak hesaplanıyor.","Herhangi iki matematik kitabının yan yana olmaması koşuluyla, önce diğer kitaplar dizilip (9 kitap) 10 araya 3 matematik kitabı yerleştirilerek çözüm yapılıyor."]},"beginTime":3575,"endTime":3836,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3575&ask_summarization=1"},{"index":22,"title":"Matematik Kitaplarının Düzenlenmesi","list":{"type":"unordered","items":["Dokuz kitabın kendi arasında sıralanması 9 faktöriyel şeklinde hesaplanır.","Üç matematik kitabı aralara yerleştirilirken, ilk matematik kitabı 10 farklı araya, ikinci matematik kitabı 9 farklı araya, üçüncü matematik kitabı ise 8 farklı araya yerleştirilebilir.","Herhangi iki matematik kitabı yan yana gelmeyecek şekilde sıralama, önce diğer kitapların sıralanması (9 faktöriyel) ve sonra 10 aradan 3'ünü seçme (10 seçme 3) ile çarpı 3 faktöriyel şeklinde hesaplanır."]},"beginTime":3837,"endTime":3913,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3837&ask_summarization=1"},{"index":23,"title":"Matematik ve Kimya Kitaplarının Sıralaması","list":{"type":"unordered","items":["En başa matematik, en sona kimya kitabı gelmesi şartıyla sıralama yapılır.","Üç matematik kitabından biri ve dört kimya kitabından biri seçildikten sonra, geriye kalan 10 kitap kendi arasında 10 faktöriyel şekilde sıralanabilir.","En başa matematik, en sona matematik gelmeyecek şekilde sıralama yapılırken, en sona ya fizik ya kimya gelecek (9 kitaptan biri) ve geriye kalan 10 kitap 10 faktöriyel şeklinde sıralanır."]},"beginTime":3913,"endTime":4056,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=3913&ask_summarization=1"},{"index":24,"title":"Fizik ve Kimya Kitaplarının Sıralaması","list":{"type":"unordered","items":["En sona fizik gelmesin, kimyalar yan yana olmak kaydıyla sıralama yapılır.","En sona kimya gelirse, kimyaların hepsi yan yana olmak zorunda olduğu için dört kimya kitabı bir araya getirilir.","En sona matematik gelirse, dört kimya kitabı bir araya getirilir ve geriye kalan kitaplar sıralanır.","En sona fizik gelmezse, kimyalar kendi arasında 4 faktöriyel, geriye kalan 8 kitap 8 faktöriyel şeklinde sıralanır veya matematik sona gelirse, dört kimya kitabı bir araya getirilir ve geriye kalan 8 kitap 8 faktöriyel şeklinde sıralanır."]},"beginTime":4056,"endTime":4206,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=4056&ask_summarization=1"},{"index":25,"title":"Oturma Düzeni Problemleri","list":{"type":"unordered","items":["Anne-baba ve dört çocuk bir banka oturacaklar, anne ile babanın arasına herhangi bir çocuk oturacak, bu durum için çözüm 3! × 2! = 12 farklı şekilde olabilir.","Anne-baba sağında olmak kaydıyla oturma düzeni için çözüm 5! = 120 farklı şekilde olabilir.","Anne-baba sağında veya solunda olmak kaydıyla oturma düzeni için çözüm 6! / 2 = 360 farklı şekilde olabilir."]},"beginTime":4211,"endTime":4368,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=4211&ask_summarization=1"},{"index":26,"title":"Çocukların Boy Sırasına Göre Oturma","list":{"type":"unordered","items":["Çocuklar soldan sağa doğru boy sırasına göre otursunlar koşuluyla oturma düzeni için çözüm 6! / 4! = 30 farklı şekilde olabilir.","Çocuklar boy sırasına göre otursunlar (soldan sağa veya sağdan sola) koşuluyla oturma düzeni için çözüm 6! × 2 / 4! = 60 farklı şekilde olabilir.","Sağ ve sol uclarda çocuklar olacak şekilde oturma düzeni için çözüm 4 × 3 × 4! = 288 farklı şekilde olabilir."]},"beginTime":4368,"endTime":4587,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=4368&ask_summarization=1"},{"index":27,"title":"Rakamları Yer Değiştirerek Sayı Oluşturma","list":{"type":"unordered","items":["3, 4, 6, 8, 9 rakamlarıyla oluşturulan beş basamaklı sayıların küçükten büyüğe doğru sıralamasında 80. sayı bulunuyor.","Rakamların yer değiştirilerek oluşturulan sayılar incelenerek 80. sayı 84.396 olarak belirleniyor.","Rakamların yer değiştirilerek oluşturulan sayıların sıralamasında 80. sayının 84.396 olduğu hesaplanıyor."]},"beginTime":4587,"endTime":4826,"href":"/video/preview/13661777911492186491?parent-reqid=1765274178701270-17590321114164735507-balancer-l7leveler-kubr-yp-sas-128-BAL&text=N1ze1xFut&t=4587&ask_summarization=1"},{"index":28,"title":"Olasılık Problemleri Çözümü","list":{"type":"unordered","items":["Eğitmen, on beş kişilik bir sınıfın ayeti 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