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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос: Сколькими способами можно выбрать 4 согласные и 2 гласные буквы в слове «безболезненный»? Сколько слов можно получить, меняя порядок букв в этом слове? Та же задача, но если после буквы "о" должна идти буква "л"? Та же задача, но если три буквы "н" не стоят радом, стоят рядом?
Чтобы уточнить этот вопрос, я отправил Вам в мини-форуме консультации следующее сообщение:
На это сообщение Вы не ответили. Поэтому я ограничусь рассмотрением простейшего случая. Для слова "безболезненный" {б, з, л, н, й} -- множество согласных, {е, о, ы} -- множество гласных букв. Тогда по формуле на с. 184 и в соответствии с правилом произведения на с. 167 [1] искомое количество способов составляет
C54*C32=5!/(4!*1!)*3!/(2!*1!)=5*3=15.
Совокупность букв слова "безболезненный" состоит из n=14 элементов девяти типов: n1=2 букв "б", n2=3 букв "е", n3=2 букв "з", n4=1 букв "о", n5=1 букв "л", n6=3 букв "н", n7=1 букв "ы", n8=1 букв "й". Согласно формуле (5) [2, с. 421], меняя порядок букв в этом слове можно получить
Если в слове после буквы "о" должна идти буква "л", то буквосочетание "ол" можно считать одной буквой. Тогда количество слов, в которых после буквы "о" идёт буква "л", равно
13!/(2!*3!*2!*1!*3!*1!*1!)=43243800.
Если в слове три буквы "н" стоят рядом, то их можно считать одной буквой. Тогда количество слов, в которых три буквы "н" стоят рядом, равно
12!/(2!*3!*2!*1!*1!*1!*1!*1!)=19958400,
а количество слов, в которых три буквы "н" не стоят рядом, равно
605404800-19958400=585446400.
Литература 1. Плотников А. Д. Дискретная математика. -- М.: Новое знание, 2006. -- 304 с. 2. Цыпкин А. Г. Справочник по математике для средних учебных заведений. -- М.: Наука, 1983. -- 483 с.
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