Evaluate each factorial expression.
106!
104!
There is no exact way to evaluate factorials except to calculate them by multiplication:
They turn out to be:
104!=102990167451456276238485838647650442830537724549990721823254917768878717324
75287174542709871683888003235965704141638377695179741979175588724736000000000000
000000000000
=1.0299016745145628*10^166 (approx.)
and
106!=114628056373470835453434738414834942870388487424139673389282723476762012382
44994625266036087184167347601629828709643514374735052822822430250631168000000000
0000000000000000
=1.1462805637347084*10^170 (approx.)
If only an approximation is required, Stirling's approximation could be used for large factorials (n>100):
n! ~ sqrt(2πn)(n/e)^n
where the error is of the order 1/n.
A closer approximation can be obtained by summing the series:
n! ~ sqrt(2πn)(n/e)^n(1+1/(12n)+1/(288n²)-139/(51840n³)-...)
For more information, see:
http://en.wikipedia.org/wiki/Stirling%27s_approximation
To evaluate factorial expressions like 106! and 104!, we need to understand what factorial means. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers from 1 to n.
For example, 5! (read as "5 factorial") is calculated as:
5! = 5 x 4 x 3 x 2 x 1 = 120
Now let's evaluate 106!:
106! = 106 x 105 x 104 x ... x 3 x 2 x 1
Since calculating factorials for such large numbers manually would be impractical, we can use the concept of logarithms to estimate the value.
However, keep in mind that the exact value of extremely large factorials like 106! is challenging to determine without specialized methods and tools. Therefore, we will use logarithmic estimation instead.
Using Stirling's approximation, we can estimate factorials as:
n! ≈ (n/e)^n √(2πn)
So for 106!, the approximation would be:
106! ≈ (106/e)^106 √(2π*106)
To calculate this approximation, we need the value of the mathematical constant e (approximately 2.71828), and √(2π) (approximately 2.50663).
106! ≈ (106/2.71828)^106 * 2.50663 * √106
Now let's evaluate 104! using the same estimation method:
104! ≈ (104/2.71828)^104 * 2.50663 * √104
Please note that these calculations are approximations, not the exact values. To obtain a more precise answer, specialized software or tools capable of handling large numbers would be required.