Percent-ages

Algebra Level 3

1 % of 2 % of 3 % ⋯ 97 % of 98 % of 99 % of 100 is equal to x ! x x − 1 1\% \ \text{of} \ 2\% \ \text{of} \ 3\% \ \cdots \ 97\% \ \text{of} \ 98\% \ \text{of} \ 99\% \ \text{of} \ 100 \text{ is equal to } \dfrac{x!}{x^{x-1}}

What is the value of x x ?

Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .

99 101 100 98

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Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
Jul 5, 2016

1 % of 2 % of 3 % ⋯ 97 % of 98 % of 99 % of 100 = 1 100 × 2 100 × 3 100 × ⋯ × 97 100 × 98 100 × 99 100 × 100 = 1 × 2 × 3 × ⋯ × 97 × 98 × 99 × 100 100 × 100 × 100 × ⋯ × 100 × 100 × 100 ⏟ 99 times = 100 ! 100 99 ∴ x = 100 1\% \ \text{of} \ 2\% \ \text{of} \ 3\% \ \cdots \ 97\% \ \text{of} \ 98\% \ \text{of} \ 99\% \ \text{of} \ 100\\ \\ = \dfrac{1}{100} × \dfrac{2}{100} × \dfrac{3}{100} × \cdots × \dfrac{97}{100} × \dfrac{98}{100} × \dfrac{99}{100} × 100\\ \\ = \dfrac{1×2×3×\cdots×97×98×99×100}{\underbrace{100×100×100×\cdots×100×100×100}_{\text{99 times}}}\\ \\ = \dfrac{100!}{{100}^{99}}\\ \\ \therefore x = \color{#3D99F6}{\boxed{100}}

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