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Консультация № 197219
27.11.2019, 10:04
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Задача по комбинаторике. Сколькими способами можно разбить 5 мужчин и 3 женщин на 2 группы по 4 человека так, чтобы в каждой группе было хотя бы по одной женщине?
Задача по комбинаторике. Сколькими способами можно разбить 5 мужчин и 3 женщин на 2 группы по 4 человека так, чтобы в каждой группе было хотя бы по одной женщине?
Обсуждение
01.12.2019, 05:29
общий
это ответ
Здравствуйте, sasha181999_9!
Задача сводится к определению того, сколькими способами можно выделить из 5 мужчин и 3 женщин группу в 4 человека, в которой будет одна или две женщины (тогда остальные 4 человека, в том числе две или одна женщины, образуют вторую группу). Всего выбрать 4 человека из 8 можно
способами. Исключим те варианты, в которых в эту группу входят 3 женщины и 1 из 5 мужчин (их, очевидно, будет 5), и те, в которых группа состоит из 4 мужчин (их также будет 5), тогда остаётся 70 - 5 - 5 = 60 способов.
Можно решить и по-другому. Для группы можно выбрать 1 из 3 женщин (3 способами) и независимо от этого 3 из 5 мужчин
способами (всего 3[$183$]10 = 30 вариантов), либо 2 женщин (также 3 способами) и 2 из 5 мужчин
способами (также 3[$183$]10 = 30 вариантов), что даёт в сумме те же 30 + 30 = 60 способов.
Задача сводится к определению того, сколькими способами можно выделить из 5 мужчин и 3 женщин группу в 4 человека, в которой будет одна или две женщины (тогда остальные 4 человека, в том числе две или одна женщины, образуют вторую группу). Всего выбрать 4 человека из 8 можно
способами. Исключим те варианты, в которых в эту группу входят 3 женщины и 1 из 5 мужчин (их, очевидно, будет 5), и те, в которых группа состоит из 4 мужчин (их также будет 5), тогда остаётся 70 - 5 - 5 = 60 способов.
Можно решить и по-другому. Для группы можно выбрать 1 из 3 женщин (3 способами) и независимо от этого 3 из 5 мужчин
способами (всего 3[$183$]10 = 30 вариантов), либо 2 женщин (также 3 способами) и 2 из 5 мужчин
способами (также 3[$183$]10 = 30 вариантов), что даёт в сумме те же 30 + 30 = 60 способов.
5
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