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В детском саду проводится новогодний праздник. У деда мороза n различных типов подарков. Каждому ребенку он может дать максимум 1 экземпляр подарка конкретного типа, при этом учитываются следующие условия:
1. Ни один ребенок не может остаться без подарка. 2. Два ребенка не могут получить одинаковый комплект подарков. 3. У каждой пары детей дожен найтись подарок одного типа.
Сколько детей можно позвать на праздник?
Уже задавал этот вопрос, но ответ "Можно позвать 2[sup]n-1[/sup] детей. Всем детям раздадим подарок типа 1. Остальным детям раздаем все подмножества из n-1 подарков - от пустого до полного. Подмножеств 2n-1. Больше позвать нельзя, так как тогда придется не давать подарок типа 1, а тогда нарушается условие 3 с ребенком, который получил только тип 1. " - не устраивает...
Обсуждение
Неизвестный
20.02.2010, 13:52
общий
Иванов Андрей Владимирович: По-моему ответ Гаряка Асмик верен. Единственный способ выполнить условие 3 в общем случае - это раздать всем детям подарок одного типа. Выполнение условия 2 достигается перебором всех подмножеств из n - 1 типа подарков, количество вариантов таких подмножеств есть 2n-1. Решение несколько несправедливо в том смысле, что кому-то дадут только один подарок, а кому-то - все n штук. Но в задаче не сказано, что у детей должно быть одинаковое число подарков.
Ну это строго не доказано. Пока не знаю, как доказать, что больше нельзя. Но вряд ли повторное задавание что-то изменит. Все эксперты вопрос уже видели... Можно спросить где-то еще.
Иванов Андрей Владимирович: ОК, пойдем по методу математической индукции. Для 2 типов подарков очевидно, что формула верна. Пусть она верна для n. Детей 2n-1 и это максимум. То есть добавить к этому комплекту нечего. Для n+1 - новый тип подарков построим множество из предыдущего так: 2n тот же комплект и еще 2n с добавлением нового вида подарка. Можно ли увеличить это число? Нет, так как это будет отсутствовавший в множестве для n либо такой же, но с подарком нового типа. В любом случае конфликт с условием 3.
Неизвестный
20.02.2010, 19:21
общий
>> Гаряка Асмик
Хорошо. Спасибо. Надеюсь, что такой ответ прокатит (задача из д/р).
Неизвестный
20.02.2010, 20:09
общий
Гаряка Асмик: Здравствуйте.
Вы в обоих вариантах предлагаете конкретный способ построения множества комплектов подарков. Таким образом Вы показываете, что можно раздать 2n-1 комплектов подарков, но никоим образом не отметается возможность построения множества большей размерности.
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