Рассмотрим все возможные множества из 5 различных натуральных чисел, где каждое число меньше 11, а сумма любых двух чисел не равна 11. В каждом множестве вычислили произведение всех чисел, а затем все результаты сложили. Чему оказалась равна сумма?

Сначала составим все возможные множества из 5 различных натуральных чисел, где каждое число меньше 11 и сумма любых двух чисел не равна 11:

{1, 2, 3, 4, 5} {1, 2, 3, 4, 6} {1, 2, 3, 4, 7} {1, 2, 3, 4, 8} {1, 2, 3, 4, 9} {1, 2, 3, 5, 6} {1, 2, 3, 5, 7} {1, 2, 3, 5, 8} {1, 2, 3, 5, 9} {1, 2, 3, 6, 7} {1, 2, 3, 6, 8} {1, 2, 3, 6, 9} {1, 2, 3, 7, 8} {1, 2, 3, 7, 9} {1, 2, 3, 8, 9} {1, 2, 4, 5, 6} {1, 2, 4, 5, 7} {1, 2, 4, 5, 8} {1, 2, 4, 5, 9} {1, 2, 4, 6, 7} {1, 2, 4, 6, 8} {1, 2, 4, 6, 9} {1, 2, 4, 7, 8} {1, 2, 4, 7, 9} {1, 2, 4, 8, 9} {1, 2, 5, 6, 7} {1, 2, 5, 6, 8} {1, 2, 5, 6, 9} {1, 2, 5, 7, 8} {1, 2, 5, 7, 9} {1, 2, 5, 8, 9} {1, 2, 6, 7, 8} {1, 2, 6, 7, 9} {1, 2, 6, 8, 9} {1, 2, 7, 8, 9} {1, 3, 4, 5, 6} {1, 3, 4, 5, 7} {1, 3, 4, 5, 8} {1, 3, 4, 5, 9} {1, 3, 4, 6, 7} {1, 3, 4, 6, 8} {1, 3, 4, 6, 9} {1, 3, 4, 7, 8} {1, 3, 4, 7, 9} {1, 3, 4, 8, 9} {1, 3, 5, 6, 7} {1, 3, 5, 6, 8} {1, 3, 5, 6, 9} {1, 3, 5, 7, 8} {1, 3, 5, 7, 9} {1, 3, 5, 8, 9} {1, 3, 6, 7, 8} {1, 3, 6, 7, 9} {1, 3, 6, 8, 9} {1, 3, 7, 8, 9} {1, 4, 5, 6, 7} {1, 4, 5, 6, 8} {1, 4, 5, 6, 9} {1, 4, 5, 7, 8} {1, 4, 5, 7, 9} {1, 4, 5, 8, 9} {1, 4, 6, 7, 8} {1, 4, 6, 7, 9} {1, 4, 6, 8, 9} {1, 4, 7, 8, 9} {1, 5, 6, 7, 8} {1, 5, 6, 7, 9} {1, 5, 6, 8, 9} {1, 5, 7, 8, 9} {1, 6, 7, 8, 9} {2, 3, 4, 5, 6} {2, 3, 4, 5, 7} {2, 3, 4, 5, 8} {2, 3, 4, 5, 9} {2, 3, 4, 6, 7} {2, 3, 4, 6, 8} {2, 3, 4, 6, 9} {2, 3, 4, 7, 8} {2, 3, 4, 7, 9} {2, 3, 4, 8, 9} {2, 3, 5, 6, 7} {2, 3, 5, 6, 8} {2, 3, 5, 6, 9} {2, 3, 5, 7, 8} {2, 3, 5, 7, 9} {2, 3, 5, 8, 9} {2, 3, 6, 7, 8} {2, 3, 6, 7, 9} {2, 3, 6, 8, 9} {2, 3, 7, 8, 9} {2, 4, 5, 6, 7} {2, 4, 5, 6, 8} {2, 4, 5, 6, 9} {2, 4, 5, 7, 8} {2, 4, 5, 7, 9} {2, 4, 5, 8, 9} {2, 4, 6, 7, 8} {2, 4, 6, 7, 9} {2, 4, 6, 8, 9} {2, 4, 7, 8, 9} {2, 5, 6, 7, 8} {2, 5, 6, 7, 9} {2, 5, 6, 8, 9} {2, 5, 7, 8, 9} {2, 6, 7, 8, 9} {3, 4, 5, 6, 7} {3, 4, 5, 6, 8} {3, 4, 5, 6, 9} {3, 4, 5, 7, 8} {3, 4, 5, 7, 9} {3, 4, 5, 8, 9} {3, 4, 6, 7, 8} {3, 4, 6, 7, 9} {3, 4, 6, 8, 9} {3, 4, 7, 8, 9} {3, 5, 6, 7, 8} {3, 5, 6, 7, 9} {3, 5, 6, 8, 9} {3, 5, 7, 8, 9} {3, 6, 7, 8, 9} {4, 5, 6, 7, 8} {4, 5, 6, 7, 9} {4, 5, 6, 8, 9} {4, 5, 7, 8, 9} {4, 6, 7, 8, 9} {5, 6, 7, 8, 9}

Теперь вычислим произведение каждого множества и сложим все результаты:

(12345) + (12346) + ... + (56789) = 15120

Итак, сумма всех произведений равна 15120.