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Answer :
The expression can be simplified to (181!) / (55!) by recognizing the pattern of factorials in the product of numbers from 181 to 56. This results in a more compact representation of the original product.
The expression 181 x 180 x 179 x .... x 58 x 57 x 56 can be simplified using the concept of factorials.
First, let's understand what a factorial is. The factorial of a number is the product of that number and all the positive integers less than it down to 1. It is denoted by an exclamation mark (!). For example, 5! (read as "5 factorial") is calculated as 5 x 4 x 3 x 2 x 1, which equals 120.
In this case, we are looking at a product of numbers from 181 down to 56. This can be written as 181 x 180 x 179 x .... x 58 x 57 x 56.
To simplify this expression using factorials, we can rewrite it as (181!) / (55!).
1. We start with the original expression: 181 x 180 x 179 x .... x 58 x 57 x 56.
2. We notice that each number in the expression is one less than the corresponding number in the factorial. For example, 180 is one less than 181, 179 is one less than 180, and so on.
3. We can rewrite the expression as (181!) / (55!). The numerator (181!) represents the factorial of 181, and the denominator (55!) represents the factorial of 55.
4. By canceling out common factors, we simplify the expression further. For example, in the numerator, the factorial of 181 includes the factors 180!, 179!, and so on. These common factors can be canceled out with the denominator's factorial terms.
5. After canceling out all the common factors, we are left with the simplified expression.
Therefore, the expression 181 x 180 x 179 x .... x 58 x 57 x 56 is equivalent to (181!) / (55!).
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