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Algebra Difficulty 4.7 AIME Find the answer

Pick a subset of at least four of the following seven numbers, order them from least to greatest, and write down their labels (corresponding letters from A through G) in that order: (A) π\pi; (B) 2+3\sqrt{2}+\sqrt{3}; (C) 10\sqrt{10}; (D) 355113\frac{355}{113}; (E) 16tan1154tan1124016 \tan ^{-1} \frac{1}{5}-4 \tan ^{-1} \frac{1}{240} (F) ln(23)\ln (23); and (G) 2e2^{\sqrt{e}}. If the ordering of the numbers you picked is correct and you picked at least 4 numbers, then your score for this problem will be (N2)(N3)(N-2)(N-3), where NN is the size of your subset; otherwise, your score is 0.

A number or a short expression. Spacing and $ signs are ignored.

Solution

We have ln(23)<2e<π<355113<16tan1154tan11240<2+3<10\ln (23)<2^{\sqrt{e}}<\pi<\frac{355}{113}<16 \tan ^{-1} \frac{1}{5}-4 \tan ^{-1} \frac{1}{240}<\sqrt{2}+\sqrt{3}<\sqrt{10}.

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