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Video, TYT sınavına hazırlık amacıyla hazırlanmış olup, ikinci bir kayıtta devam edeceği belirtilmektedir."]},"endTime":1577,"title":"Faktöriyel Konu Anlatımı ve Problemler","beginTime":0}],"fullResult":[{"index":0,"title":"Faktöriyel Konu Anlatımı Giriş","list":{"type":"unordered","items":["Kanalın en eski videosundaki faktöriyel konu anlatımı videosu, Instagram'dan gelen bir mesaj üzerine daha iyi ve dinlenebilir bir versiyon olarak yeniden hazırlanıyor.","Faktöriyel konu anlatımı, permütasyon, kombinasyon ve olasılık konularını daha iyi anlamak için altyapı oluşturacak.","Video, faktöriyel konusunu sıfırdan zirveye kadar anlatarak öğrencileri zirvede bırakacak."]},"beginTime":10,"endTime":130,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=10&ask_summarization=1"},{"index":1,"title":"Faktöriyel Tanımı","list":{"type":"unordered","items":["Faktöriyel, pozitif tam sayılar kümesinden alınan bir sayıyı, o sayıdan bir'e kadar olan tüm sayıları çarpma işlemidir.","Faktöriyel, bir çarpım sembolü olarak da tanımlanabilir ve alt kısımdan başlayıp üst kısıma kadar olan sayılar çarpılır.","Faktöriyel, bir sayıdan bir'e kadar olan tüm tam sayıların çarpımı olarak da ifade edilebilir."]},"beginTime":130,"endTime":216,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=130&ask_summarization=1"},{"index":2,"title":"Faktöriyel Örnekleri","list":{"type":"unordered","items":["1! = 1, 2! = 1×2 = 2, 3! = 1×2×3 = 6, 4! = 1×2×3×4 = 24 şeklinde hesaplanır.","Faktöriyel değerleri arasında bir ilişki vardır: n! = (n-1)! × n şeklinde ifade edilebilir.","Faktöriyel değerleri büyüdükçe artar ve k! ≤ n! olmak zorundadır (k ≤ n)."]},"beginTime":216,"endTime":353,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=216&ask_summarization=1"},{"index":3,"title":"Sıfır Faktöriyel","list":{"type":"unordered","items":["0! = 1 olarak tanımlanır, bu durum faktöriyel işlemlerinin tutarlılığını sağlar.","n! = (n-1)! × n formülüne göre 1! = 0! × 1 = 1 olur, bu da 0! = 1 olduğunu gösterir.","Faktöriyel fonksiyonu tanımı gereği de 0! = 1 olarak alınır."]},"beginTime":353,"endTime":422,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=353&ask_summarization=1"},{"index":4,"title":"Faktöriyel Özellikleri","list":{"type":"unordered","items":["Faktöriyelde, birbirine eşit olmayan pozitif tam sayılar için, daha küçük sayıların faktöriyeli de daha küçük olacaktır.","Faktöriyel \"sıra korur\", yani a \u003c b ise a! \u003c b! olacaktır.","Sadece 0! ve 1! tek sayıdır, 2! ve sonrası tüm faktöriyeller çift sayıdır çünkü en az bir tane çift sayı içerir."]},"beginTime":425,"endTime":515,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=425&ask_summarization=1"},{"index":5,"title":"Faktöriyel Açılımı ve Özellikleri","list":{"type":"unordered","items":["n! açılımında k! bulunur çünkü n! = n × (n-1) × ... × k × (k-1) × ... × 1 şeklinde açılır.","Faktöriyel sorularında, payda eşitleme yaparak sadeleştirme yapılabilir.","Faktöriyel sorularında değer koyarak çözüm yapmak, denklem çözmekten daha kolay olabilir."]},"beginTime":515,"endTime":813,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=515&ask_summarization=1"},{"index":6,"title":"Faktöriyel Denklemleri","list":{"type":"unordered","items":["Faktöriyel içeren denklemlerde, faktöriyel ifadelerini sadeleştirerek çarpanlara ayırma yapabiliriz.","Faktöriyel içeren denklemlerde, faktöriyel ifadelerini açarak ve değer koyarak çözüm yapılabilir.","Faktöriyel sorularında, faktöriyel ifadelerini sadeleştirerek ve çarpanlara ayırarak çözüm yapılabilir."]},"beginTime":813,"endTime":1024,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=813&ask_summarization=1"},{"index":7,"title":"Faktöriyel İşlemleri","list":{"type":"unordered","items":["Faktöriyel hesaplamalarında çarpım işlemlerinde, bir ile çarpma işlemi yapmanın etkisiz olduğu gösteriliyor.","50 faktöriyel bölü 18 faktöriyel şeklinde bir sonuç elde ediliyor.","Faktöriyel işlemlerinde, 20 faktöriyel ile 19 faktöriyel arasındaki ilişki gösteriliyor."]},"beginTime":1027,"endTime":1114,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1027&ask_summarization=1"},{"index":8,"title":"Faktöriyel Problemleri","list":{"type":"unordered","items":["20 faktöriyel ile 19 faktöriyel arasındaki ilişkiyi bulmak için 20 faktöriyel = 20 × 19 faktöriyel şeklinde ifade ediliyor.","Faktöriyel problemlerinde, sayıların toplamının faktöriyel şeklinde yazılması gösteriliyor.","120 faktöriyel = n faktöriyel şeklinde bir denklem kurulup, n = 6 olarak bulunuyor."]},"beginTime":1114,"endTime":1282,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1114&ask_summarization=1"},{"index":9,"title":"Faktöriyel Özellikleri","list":{"type":"unordered","items":["5 faktöriyel ve sonrası olan faktöriyel sayılarının son basamağının sıfır olduğu belirtiliyor.","1'den 4'e kadar faktöriyel sayılarının toplamının 33 olduğu gösteriliyor.","Büyük faktöriyel sayılarının son basamaklarının sıfır olduğu, bu nedenle toplamda etkisiz kaldığı açıklanıyor."]},"beginTime":1282,"endTime":1414,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1282&ask_summarization=1"},{"index":10,"title":"Mod ve Kuvvet İşlemleri","list":{"type":"unordered","items":["3'ün 123 kuvvetinin son basamağını bulmak için mod işlemi kullanılıyor.","3'ün kuvvetleri incelendiğinde dördüncü kuvvette bir döngü olduğu gösteriliyor.","3'ün 123 kuvvetinin son basamağının 7 olduğu sonucuna varılıyor."]},"beginTime":1414,"endTime":1528,"href":"/video/preview/3614844874140148965?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1414&ask_summarization=1"},{"index":11,"title":"Faktöriyel 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İçerik, faktöriyelin asal çarpanları, bölme mantığı, faktöriyel sayılarının son basamakları, ardışık faktöriyel ifadelerinde parantez alma ve faktöriyel içindeki çarpanları bulma gibi konuları kapsamaktadır.","Video, test hazırlığı için tekrar yapma ve sınava yakın soruları çözmek amacıyla hazırlanmış olup, faktöriyel konusundaki temel kavramları pekiştirmek isteyenler için faydalı olabilir. Son bölümde bir kutu içine yerleştirilen sayıların çarpımlarını içeren faktöriyel hesaplaması sorusu çözülmekte ve bir sonraki testte sadece yeni nesil soruların olacağı bilgisi verilmektedir."]},"endTime":1930,"title":"Faktöriyel Konusu Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Faktöriyel Konusuna Giriş","list":{"type":"unordered","items":["Miray Yayınları TYT soru bankasında faktöriyel konusu üç testten oluşmaktadır.","İlk testte hem konuyu hatırlatacak hem de yeni nesil sorulara geçiş sağlayacak sorular bulunmaktadır.","Faktöriyel, permütasyon, kombinasyon ve binom konularında kullanılan bir kavramdır ve son yıllarda TYT ve AYT sınavlarında pek soru gelmeyen bir konudur."]},"beginTime":8,"endTime":108,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=8&ask_summarization=1"},{"index":1,"title":"Faktöriyel Sorularının Çözümü","list":{"type":"unordered","items":["Test 1'in 7. sorusunda a ve b doğal sayılar olmak üzere a!/(b! + b!) = 121 eşitliğini sağlayan 4 farklı (a,b) sıralı ikilisi vardır.","Test 1'in 8. sorusunda iki basamaklı a ve pozitif tam sayılar için a! = 5 × 5! × b! şartını sağlayan a+b toplamının en az değeri 48'dir.","Test 1'in 11. sorusunda en! sayısının son sekiz basamağı sıfır olduğuna göre, en'in en küçük değeri 35, en büyük değeri 39'dur ve toplamları 74'tür."]},"beginTime":108,"endTime":442,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=108&ask_summarization=1"},{"index":2,"title":"Faktöriyel Sorularının Devamı","list":{"type":"unordered","items":["Test 1'in 16. sorusunda x² = 81 olduğuna göre, pozitif tam sayı x değeri 9'dur.","Test 2'nin 3. sorusunda a doğal sayı olmak üzere a! sayısının altı farklı asal çarpanı olduğuna göre, a'nın alabileceği farklı değer sayısı 4'tür."]},"beginTime":442,"endTime":638,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=442&ask_summarization=1"},{"index":3,"title":"Faktöriyel Problemi Çözümü","list":{"type":"unordered","items":["Soruda b çift sayı olduğu belirtilmiş ve a'nın kaç farklı değer alabileceği sorulmuş.","Ardışık faktöriyel ifadelerinde parantez alınarak 12×10! / 2²a şeklinde düzenlenmiş.","B'nin çift sayı olması için a'nın 0, 1, 2, 3, 4 değerlerini alabileceği bulunmuş, toplam 5 farklı değer olduğu sonucuna varılmış."]},"beginTime":644,"endTime":799,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=644&ask_summarization=1"},{"index":4,"title":"Faktöriyel İfadelerinin Düzenlenmesi","list":{"type":"unordered","items":["Ardışık faktöriyel ifadelerinde parantez alınarak 4!×5!×6! / n!³ şeklinde düzenlenmiş.","n'nin 0, 1, 2, 3, 4, 5 değerlerini alabileceği bulunmuş, toplam 6 farklı değer olduğu sonucuna varılmış."]},"beginTime":799,"endTime":977,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=799&ask_summarization=1"},{"index":5,"title":"Faktöriyel ve Üslü Sayılar Problemi","list":{"type":"unordered","items":["a ve b pozitif tam sayılar olmak üzere 4, 8, 12, ..., 64 sayılarının çarpımı 2^a × b! şeklinde ifade edilmiş.","4'ün 16 faktöriyel ile çarpımı ve 16 faktöriyelin 2^32 şeklinde yazılması sonucu a=36 ve b=15 bulunmuş.","a+b toplamının alabileceği en büyük değer 51 olarak hesaplanmış."]},"beginTime":977,"endTime":1145,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=977&ask_summarization=1"},{"index":6,"title":"Sözel ve Faktöriyel Problemi","list":{"type":"unordered","items":["Anıl ve Kaan'ın sırasıyla x ve y tane bilyeleri olduğu belirtilmiş.","Anıl bilyelerinin sayısının 10'dan çıkarılıp faktöriyeli, Kaan bilyelerinin sayısının 3 katından 12 çıkarılıp faktöriyeli alınmış.","Bu işlemler sonunda bulunan sayıların birbirine eşit olduğu ve bilye sayıları toplamının en çok kaç olabileceği sorulmuş."]},"beginTime":1145,"endTime":1192,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1145&ask_summarization=1"},{"index":7,"title":"Faktöriyel Problemi Çözümü","list":{"type":"unordered","items":["Faktöriyelin tanımı gereği, faktöriyel ile başlayan sayı sıfırdan büyük veya eşit olmalıdır.","x ve y sayılarının faktöriyel değerlerinin toplamı en çok olabilmesi için x=10 ve y=4 seçilmelidir.","Bu durumda x+y+1 faktöriyel toplamı 8 olur ve sorunun cevabı Bursa'dır."]},"beginTime":1199,"endTime":1287,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1199&ask_summarization=1"},{"index":8,"title":"Özel İşlem Problemi","list":{"type":"unordered","items":["Verilen özel işlemde n+1 faktöriyel bölü n-1 faktöriyel ifadesi 42'ye eşittir.","İşlem sonucunda n=6 olarak bulunur ve sorunun cevabı Bursa'dır."]},"beginTime":1287,"endTime":1374,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1287&ask_summarization=1"},{"index":9,"title":"Faktöriyel ve Ok Problemi","list":{"type":"unordered","items":["İki ok aşağıya doğru baktığında oluşan sayının faktöriyeli alınır, mavi ok aşağıya, kırmızı ok yukarı baktığında faktöriyel alınır.","İki ok yukarı doğru baktığında kırmızı dört'ü c'ye bölüp faktöriyel alınır ve bulunan sayının bir daha faktöriyel alınır.","Koşulunu sağlayan dört tane sayı vardır ve sorunun cevabı Ceyhan'dır."]},"beginTime":1374,"endTime":1563,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1374&ask_summarization=1"},{"index":10,"title":"Rakamların Toplamı Problemi","list":{"type":"unordered","items":["Kutulara yazılan rakamların toplamı 16'dır ve bu rakamların çarpımı bir sayının faktöriyelidir.","Rakamlar 1, 3, 2, 6 ve 5 olarak seçildiğinde toplamları 16 olur ve çarpımı 5 faktöriyel olur.","a sayısı 5'e eşittir."]},"beginTime":1563,"endTime":1684,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1563&ask_summarization=1"},{"index":11,"title":"Faktöriyel Dizisi Problemi","list":{"type":"unordered","items":["Belirli bir kurala göre oluşturulan sayı dizisinde basamak değerleri verilmiştir.","Basamak değerleri her seferinde bir fazlası kadar artmaktadır.","Son altı basamağı sıfır olan faktöriyel 29 faktöriyeldir ve sorunun cevabı 7'dir."]},"beginTime":1684,"endTime":1793,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1684&ask_summarization=1"},{"index":12,"title":"Matematik Probleminin Çözümü","list":{"type":"unordered","items":["Kutunun içerisine eni yerleştirildiğinde, sağ tarafta 2'den 16'ya kadar olan çift sayıların çarpımı, sol tarafta ise 1'den 17'ye kadar olan tek sayıların çarpımı elde edilir.","Kutu içerisindeki sayı faktöriyel olarak ifade edilir: 16! / 15!","İşlem sonucunda 16! parantezine alınarak sadeleştirme yapılır ve sonuç 16² = 2⁸ olarak bulunur."]},"beginTime":1795,"endTime":1914,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1795&ask_summarization=1"},{"index":13,"title":"Test Hakkında Bilgi","list":{"type":"unordered","items":["İlk iki testte zorlayıcı sorular çözülmüştür.","Sonraki testte sadece yeni nesil sorular olacak ve toplam dokuz soru bulunmaktadır.","Eğitmen, izleyicilere çalışmayı bırakmamalarını tavsiye ederek vedalaşıyor."]},"beginTime":1914,"endTime":1929,"href":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=1914&ask_summarization=1"}],"linkTemplate":"/video/preview/13940319673926856557?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Factorial 1 - now IT'S the new 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İçerik, faktöriyel sayılarının asal çarpanlarını bulma, faktöriyel sayılarının sonunda kaç tane sıfır olduğunu hesaplama, faktöriyel sayılarının bölen en büyük asal sayısını bulma ve faktöriyel sayılarının bölünebilme özellikleri gibi konuları kapsamaktadır. Eğitmen, toplam beş farklı soru üzerinden konuyu pekiştirmektedir.","Video ayrıca 48! sayısının son basamaklarını bulma ve 25! + 26! sayısının 24! ile bölümünden kalanı hesaplama gibi pratik problemleri de içermektedir. Eğitmen, permütasyon, kombinasyon ve olasılık konularının da gelecekte anlatılacağını belirtmektedir."]},"endTime":1554,"title":"Faktöriyel ve Asal Çarpanlar Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Altı ile Bölünebilme Kuralı","list":{"type":"unordered","items":["Altı ile bölünebilmek için hem iki hem üç'e bölünmek gerekir.","Altı eşittir iki çarpı üç olduğundan, verilen ifadelerin her biri iki'yi kendi içerisinde içerdiği için ikiye tam bölünür.","Üç faktöriyel çarpı dört faktöriyel ifadesinde dört faktöriyelin sonrası da içerisinde üç barındırır, dolayısıyla altı'ya bölümünden kalan sıfırdır."]},"beginTime":12,"endTime":145,"href":"/video/preview/11523363563126338543?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=12&ask_summarization=1"},{"index":1,"title":"Faktöriyel ve Asal Çarpanlar","list":{"type":"unordered","items":["Bir sayının faktöriyelini alırken, o sayının en büyük asal çarpanı 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asal çarpanlarına ayrılır, asal çarpanların üsleri bir arttırılıp çarpılır.","15 faktöriyel sayısının pozitif tam bölenleri sayısı 234322'dir.","Eğer pozitif demeseydi, tam bölenleri sayısı deseydi, bu sonuç 4'ün iki katı olan 4656'ya eşit olurdu."]},"beginTime":645,"endTime":764,"href":"/video/preview/11523363563126338543?parent-reqid=1769307739569390-18091347434668084358-balancer-l7leveler-kubr-yp-klg-59-BAL&text=Factorial&t=645&ask_summarization=1"},{"index":5,"title":"Faktöriyel Toplamında Sıfır Sayısı","list":{"type":"unordered","items":["23 faktöriyel artı 72 faktöriyel toplamının sonunda kaç tane sıfır olacağı sorulduğunda, rol belirleyen küçük olan faktöriyeldir.","23 faktöriyel içinde 4 tane 10 üzeri 4 tane sıfır vardır.","72 faktöriyel içinde en az 14 tane sıfır vardır, bu nedenle toplamın sonunda 4 tane sıfır 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