Let's simplify the expression step-by-step.
The expression is: \[ \frac{1}{8} \times 3 \times 11 \times 6 \times 7 \times \frac{4}{5} \times \frac{15}{21} \]
First, let's break it down:
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Start with \(\frac{1}{8} \times 3 = \frac{3}{8}\).
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Now multiply by 11: \[ \frac{3}{8} \times 11 = \frac{33}{8} \]
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Next, multiply by 6: \[ \frac{33}{8} \times 6 = \frac{198}{8} = \frac{99}{4} \]
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Then multiply by 7: \[ \frac{99}{4} \times 7 = \frac{693}{4} \]
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Now multiply by \(\frac{4}{5}\): \[ \frac{693}{4} \times \frac{4}{5} = \frac{693 \times 4}{4 \times 5} = \frac{693}{5} \]
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Finally, multiply by \(\frac{15}{21}\): \[ \frac{693}{5} \times \frac{15}{21} = \frac{693 \times 15}{5 \times 21} \] Simplifying this:
- \(5\) and \(15\) can be reduced: \(15 ÷ 5 = 3\).
- \(21\) and \(693\): \(693 ÷ 21 = 33\).
Thus: \[ \frac{33 \times 3}{1} = 99 \]
The final result of the expression is: \[ \boxed{99} \]