Complete the following proof. It is not possible to arrange the numbers 1, 2, 3, …, 10 in a circle so that every triple of consecutively placed numbers has a sum less than 15. Proof: In any arrangement of 1, 2, 3, …, 10 in a circle, there _____ triples of consecutively placed numbers, because _____.Each number appears in _____ of these triples. If the sum of each triple were less than 15, then the total sum of all triples would be less than _____ times 15 or _____ But 1 + 2 + 3 + … + 10 is 55 and since each number appears in _____ triples, the total sum should be _____ times 55. This is a contradiction so not all triples can have a sum less than 15.
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