x(1-1%2F11)x(1-1%2F10)x(1-1%2F9)x(1-1%2F8).jpeg "(1-1/12)x(1-1/11)x(1-1/10)x(1-1/9)x(1-1/8)")
Unraveling the Mystery: (1-1/12)x(1-1/11)x(1-1/10)x(1-1/9)x(1-1/8)
At first glance, the expression (1-1/12)x(1-1/11)x(1-1/10)x(1-1/9)x(1-1/8) might seem intimidating. However, there's a hidden beauty and simplicity within this seemingly complex multiplication. Let's break it down and reveal the elegant pattern it holds.
Simplifying the Expression
Each term in the expression can be rewritten as a fraction:
- (1-1/12) = 11/12
- (1-1/11) = 10/11
- (1-1/10) = 9/10
- (1-1/9) = 8/9
- (1-1/8) = 7/8
Now, our expression becomes: (11/12) x (10/11) x (9/10) x (8/9) x (7/8)
The Magic of Cancellation
Notice something interesting? We have a series of consecutive numbers in the numerator and denominator. This allows for a beautiful cancellation:
- 11 in the numerator cancels with 11 in the denominator.
- 10 in the numerator cancels with 10 in the denominator.
- 9 in the numerator cancels with 9 in the denominator.
- 8 in the numerator cancels with 8 in the denominator.
This leaves us with: (11/12) x (7/8)
Final Calculation
Finally, we perform the remaining multiplication:
(11/12) x (7/8) = 77/96
The Significance
The beauty of this expression lies in its simplicity after simplification. It demonstrates a powerful pattern of cancellation that reveals the hidden elegance of mathematics. This pattern is not just a mathematical trick; it is a fundamental principle that applies to various areas, including probability and other calculations involving fractions.