HMMT February

Number theory Difficulty 4.7 AIME Find the answer United States

Problem:
Let $N$ denote the sum of the decimal digits of $\binom{1000}{100}$ . Estimate the value of $N$ . If your answer is a positive integer $A$ written fully in decimal notation (for example, 521495223), your score will be the greatest integer not exceeding $25 \cdot (0.99)^{|A-N|}$ . Otherwise, your score will be zero.

Your answer

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

Solution

Solution:
Answer: 621 http://www.wolframalpha.com/input/?i=sum+of+digits+of+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: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.