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Probability with sum of digits (Posted on 2025-06-02)

A box contains 900 cards numbered from 100 to 999. Cards are drawn randomly, one at a time, without replacement, and the sum of their digits is recorded. What is the minimum number of cards that must be drawn to guarantee that at least three of these sums are the same?

No Solution Yet Submitted by Danish Ahmed Khan

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| Comment 1 of 2

The sod of a 3-digit number can be from 1 to 27.

There is only one instance each for sod = 1 or sod = 27.

All other sods from 2 to 26 have at least 2 cards.

In the worst case, the first 52 cards could be drawn and still have no sod appear thrice: 100, 999, and 2 each for the 25 sods from 2 to 26.

So 53 is the minimum number in order to assure some sod appears more than 2 times.

Posted by Larry on 2025-06-02 09:01:02

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