Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Find the answer

Let $N$ denote the sum of the decimal digits of $\binom{1000}{100}$ . Estimate the value of $N$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

http://www.wolframalpha.com/input/?i=sum+of+digits+of $+\mathrm{nCr}(1000,100)$ . To see this, one can estimate there are about 150 digits, and we expect the digits to be roughly random, for $150 \cdot 4.5 \approx 675$ , which is already very close to the actual answer. The actual number of digits is 140, and here $140 \cdot 4.5=630$ is within 9 of the actual answer.

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.