The exact answer to the question "What is 1 2 3 4 5 all the way to 100?" is 100 factorial (100!), which is an incredibly large number with 158 digits.
Understanding the Factorial (100!)
The expression "1 2 3 4 5 all the way to 100" represents the mathematical operation known as a factorial. A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.
For n = 100, the value of 100! is:
93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000
Key Characteristics of Factorials:
- Rapid Growth: Factorials grow extremely quickly. Even small numbers result in surprisingly large products.
- Combinatorics and Probability: Factorials are fundamental in various mathematical fields, particularly in combinatorics (counting permutations and combinations) and probability theory. For example, 100! represents the number of ways 100 distinct items can be arranged.
- Trailing Zeros: The number of trailing zeros in a factorial is determined by the number of times 10 is a factor, which in turn depends on the number of times 5 and 2 are factors. For 100!, there are 24 trailing zeros.
To illustrate the rapid growth of factorials, consider this small table:
| n | n! (Factorial) | Calculation |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 1 × 2 |
| 3 | 6 | 1 × 2 × 3 |
| 4 | 24 | 1 × 2 × 3 × 4 |
| 5 | 120 | 1 × 2 × 3 × 4 × 5 |
| 10 | 3,628,800 | 1 × ... × 10 |
As you can see, 10! is already over 3.6 million, giving a sense of how vast 100! becomes. Due to their immense size, calculating large factorials typically requires computational tools or specialized algorithms rather than manual calculation.
Distinguishing Between Multiplication and Summation: A Note on the Reference
It's important to differentiate the factorial calculation (a product) from other mathematical operations like summation. The provided reference pertains to a distinct problem: "What is 1+2-3-4+5… to 100?". According to the Quora discussion, the answer to that specific summation problem is stated as 5050.
This value, 5050, is famously the sum of the first 100 positive integers (1 + 2 + 3 + ... + 100), calculated using the formula n(n+1)/2, where n=100. So, 100 (100+1) / 2 = 100 101 / 2 = 5050. While the reference notes that this particular sum can be "done in your head far faster than you can write a program to do it," this ease of calculation and the resulting value of 5050 apply to the sum of numbers, not their product as requested in your question.
Therefore, while 5050 is the answer to a related summation problem involving numbers up to 100, it is not the answer to 1 2 3 ... 100 (the factorial). The factorial operation yields an exponentially larger result.
