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Number theory Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

Run Ze writes down all the integers with the following properties: the first digit is $1$ ; each of the following digits is at least as large as the digit before it; the sum of the digits of the integer is $5$ . How many integers does he write?

(A) $9$ (B) $8$ (C) $7$ (D) $6$ (E) $5$

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