The Value of 100! to 110! Factorial Problem

  • Thread starter Cosmos
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    Factorial
In summary: Just put in the sum:110! - 109! + 108! - 107! + 106! - 105! + 104! - 103! + 102! - 101! + 100!and it will give you the result:17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000So, in summary, the value of 100!-101!+102!-103!...-109!+110! is a very large number,
  • #1
Cosmos
18
1
What do you think is the value of
100!-101!+102!-103!...-109!+110!
:biggrin:
 
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  • #2
n! = (n-1)!n. Take 100! as a common factor and go from there...
 
  • #3
Come on man! i tried ... doesn't work...
 
  • #4
Cosmos said:
Come on man! i tried ... doesn't work...
Show us that you tried...
 
  • #5
when you take out 100! common out...then you are left with (1-101+102 times 101...) which afterwards...don't know man...addition is not the way out as it gives you a very humongous number...
 
  • #6
Cosmos said:
What do you think is the value of
100!-101!+102!-103!...-109!+110!
:biggrin:
It's going to be pretty large.

Write the sum this way:

S = 110! - 109! + 108! - 107! + 106! - 105! + 104! - 103! + 102! - 101! + 100!

which can be grouped:

S = (110! - 109!) + (108! - 107!) + ... + (102! - 101!) + 100!

Now, take the term (110! - 109!) = (110 * 109! - 109!) = (110 - 1) * 109! = 109 * 109!

You can telescope the other terms in this sum in a similar fashion.

S = 109 * 109! + 107 * 107! + 105 * 105! + 103 * 103! + 101 * 101! + 100!

You can manipulate the terms in the sum above in a similar manner, but the result is clear:

S is a pretty big number no matter how you slice it.

Were you thinking that S would not be such a large number?
 
  • #7
Does it help any if you take out 110! as a factor?
 
  • #9
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
 
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  • #10
micromass said:
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
Wow! And I was tempted to answer simply O(1). However, Stirling gave me 1.58...for 110! Would be interesting to know whether the calc.exe isn't precise enough or the margin in Stirling's formula is larger than I thought.
 
  • #11
fresh_42 said:
Wow! And I was tempted to answer simply O(1). However, Stirling gave me 1.58...for 110! Would be interesting to know whether the calc.exe isn't precise enough or the margin in Stirling's formula is larger than I thought.

According to the program I just wrote:
110!=15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
So Stirling definitely is accurate here.
 
  • #12
micromass said:
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
Not sure what this number is.
 
  • #13
Using Mathematica:
In[2]:= 110! - 109! + 108! - 107! + 106! - 105! + 104! - 103! + 102! - 101! + 100!

Out[2]= 15739381947081460468710896569033260448048487750802968746988405111340773775128510600810783940010370922688077274739713895911222137779156961431310006359162880000000000000000000000000
You can do it yourself using WolframAlpha.
 

Related to The Value of 100! to 110! Factorial Problem

1. What is the value of 100! to 110! factorial problem?

The value of 100! to 110! factorial problem refers to the calculation of the factorial of numbers from 100 to 110, which is the product of all the numbers from 1 to the given number. In this case, it would be the product of 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, and 110.

2. Why is the value of 100! to 110! factorial problem important?

The value of 100! to 110! factorial problem is important because it allows us to understand the concept of factorials and their calculations. It also has practical applications in mathematics, statistics, and computer science.

3. How do you solve the value of 100! to 110! factorial problem?

To solve this problem, you would first calculate the factorial of each number from 100 to 110, and then multiply all the results together. This can be done manually or by using a calculator or computer program.

4. Can the value of 100! to 110! factorial problem be expressed in scientific notation?

Yes, the value of 100! to 110! factorial problem can be expressed in scientific notation, which is a way of writing very large or very small numbers. For example, the value of 100! to 110! factorial problem can be written as 1.0333148 x 10^172.

5. What is the approximate value of 100! to 110! factorial problem?

The approximate value of 100! to 110! factorial problem is 1.0333148 x 10^172, which is an extremely large number. This can also be written as 103,331,484,832,020,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

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