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4

Part of 2017 Czech And Slovak Olympiad III A

Problems(1)

number sections of a number with only 0s and 1s, total sum wanted

Source: Czech and Slovak Olympiad 2017, National Round, III A p4

2/2/2020
For each sequence of nn zeros and nn units, we assign a number that is a number sections of the same digits in it. (For example, sequence 0011100100111001 has 44 such sections 00,111,00,100, 111,00, 1.) For a given nn we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to (n+1)(2nn)(n+1){2n \choose n}
combinatoricsSumProbabilistic Method