Consider a string of n 1’s. We wish to place some + signs in between so that the sum is 1000. For in

Question

Consider a string of n 1’s. We wish to place some + signs in between so that the sum is 1000. For instance, if n = 190, one may put + signs so as to get 11 ninety times and 1 ten times, and get the sum 1000. If a is the number of positive integers n for which it is possible to place + signs so as to get the sum 1000, then find the sum of the digits of a.

Accepted Answer

{"content":[{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 1:","type":"text"},{"text":"  1000 = 1a1 + 11a2 + 11a3 + ……","type":"text"}]},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 2:"},{"text":" where a1, a2, a3, …… are non-negative integers for all ai, when i > 3, ai = 0","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3:","type":"text"},{"text":" => 1000 = 111p + 11q + r","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4:","type":"text"},{"text":" If ","type":"text"},{"attrs":{"latex":"p"},"type":"math_inline"},{"text":" = 0, then there are 91 possibilities for q, r.","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 5:","type":"text","marks":[{"type":"bold"}]},{"text":" For ","type":"text"},{"attrs":{"latex":"p"},"type":"math_inline"},{"type":"text","text":" = 1, there are 81 possibilities for q, r."}],"type":"paragraph"},{"content":[{"text":"Step 6:","type":"text","marks":[{"type":"bold"}]},{"text":" For ","type":"text"},{"attrs":{"latex":"p"},"type":"math_inline"},{"text":" = 2, there are 71 possibilities for q, r.","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 7:","type":"text"},{"type":"text","text":" For "},{"attrs":{"latex":"p"},"type":"math_inline"},{"type":"text","text":" = 3, there are 61 possibilities for q, r."}],"type":"paragraph"},{"content":[{"text":"Step 8:","type":"text","marks":[{"type":"bold"}]},{"text":" For ","type":"text"},{"attrs":{"latex":"p"},"type":"math_inline"},{"text":" = 4, there are 51 possibilities for q, r.","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 9:","type":"text"},{"text":" …………………………………………………","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 10:","type":"text"},{"text":" …………………………………………………","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 11:","type":"text","marks":[{"type":"bold"}]},{"text":" …………………………………………………","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 12:","type":"text"},{"text":" For ","type":"text"},{"attrs":{"latex":"p"},"type":"math_inline"},{"type":"text","text":" = 9, there are 1 possibility for q, r."}]},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 13:"},{"type":"text","text":" Hence, total number of possibilities = 460"}]},{"content":[{"text":"Step 14:","type":"text","marks":[{"type":"bold"}]},{"text":" Sum of digits of 460 is 10.","type":"text"}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

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