Instruction

1

To find the factorial of a number, you need to calculate the product of all the integers in the interval from 1 to a given number. The General formula is as follows:

n! = 1*2*...*n, where n is any nonnegative integer. A factorial is denoted with an exclamation mark.

n! = 1*2*...*n, where n is any nonnegative integer. A factorial is denoted with an exclamation mark.

2

The basic properties of factorials:

• 0! = 1;

• n! = n*(n-1)! ;

• n!^2 ≥ n^n ≥ n! ≥ n.

The second property of the factorial is called recursion and the factorial – elementary recursive function. Recursive functions are often used in the theory of algorithms and writing computer programs, since many algorithms, and programming functions have a recursive structure.

• 0! = 1;

• n! = n*(n-1)! ;

• n!^2 ≥ n^n ≥ n! ≥ n.

The second property of the factorial is called recursion and the factorial – elementary recursive function. Recursive functions are often used in the theory of algorithms and writing computer programs, since many algorithms, and programming functions have a recursive structure.

3

To determine the factorial of a large number by the Stirling formula, which gives, however, approximate equality, but with a little error. Full formula is as follows:

n! = (n/e)^n*√(2*π*n)*(1 + 1/(12*n) + 1/(288*n^2) + ...)

ln (n!) = (n + 1/2)*ln n – n + ln √(2*π),

where e is the base of natural logarithm, Euler's number, whose numerical value is made approximately equal 2,71828...; π – mathematical constant whose value was equal to 3.14.

Widespread use of the Stirling formula in the form:

n! ≈ √(2*π*n)*(n/e)^n.

n! = (n/e)^n*√(2*π*n)*(1 + 1/(12*n) + 1/(288*n^2) + ...)

ln (n!) = (n + 1/2)*ln n – n + ln √(2*π),

where e is the base of natural logarithm, Euler's number, whose numerical value is made approximately equal 2,71828...; π – mathematical constant whose value was equal to 3.14.

Widespread use of the Stirling formula in the form:

n! ≈ √(2*π*n)*(n/e)^n.

4

There are various generalizations of the notion of factorial, for example, double -, m-fold, decreasing, increasing, primorial, superfactory. The double factorial is denoted by !! and is the product of all natural numbers in the interval from 1 to the number having the same parity, for example, 6!! = 2*4*6.

5

m-fold factorial General case of the double factorial for any integer non-negative numbers m:

for n = mk – r true n!...!! = ∏ (m*I - r), where r is the set of integers from 0 to m-1, I – belongs to the set of integers from 1 to k.

for n = mk – r true n!...!! = ∏ (m*I - r), where r is the set of integers from 0 to m-1, I – belongs to the set of integers from 1 to k.

6

Decreasing factorial is written as follows:

(n)_k = n!/(n - k)!

Growing:

(n)^k = (n + k -1)!/(n - 1)!

(n)_k = n!/(n - k)!

Growing:

(n)^k = (n + k -1)!/(n - 1)!

7

Primorial number equal to the product of primes less than the number and is denoted by #, for example:

12# = 2*3*5*7*11, obviously, 13# = 11# = 12#.

Superbacteria equal to the product of the factorials of the integers in the interval from 1 to the original number, i.e.:

sf(n) = 1!*2!*3*...(n - 1)!*n! for example, sf(3) = 1!*2!*3! = 1*1*2*1*2*3 = 12.

12# = 2*3*5*7*11, obviously, 13# = 11# = 12#.

Superbacteria equal to the product of the factorials of the integers in the interval from 1 to the original number, i.e.:

sf(n) = 1!*2!*3*...(n - 1)!*n! for example, sf(3) = 1!*2!*3! = 1*1*2*1*2*3 = 12.

# Advice 2 : How to solve factorial

**The factorial**of any number is the product of all non-negative integers to define inclusive). His symbol is an exclamation point following the determined number (for example, 5!).

You will need

- Calculator

Instruction

1

In order to calculate the factorial of a number n, you need to use a simple formula: n! = 1 x 2 ... x n. For example, 5! = 1 x 2 x 3 x 4 x 5 = 120. All multiplication operations can be easily done by using the calculator.

2

If the number, whose factorial we need to calculate is too large, then you will be able to help a table of factorials. Such tables are in the printed version and free online.

3

Another option to compute the factorial for the lazy, the so – called calculator factorials, outstanding need the answer after carrying out just one operation. This calculator also is in the public domain.

# Advice 3 : How to calculate a factorial

**The factorial**of a natural number is the product of all previous natural numbers, including the number itself.

**The factorial**of zero equal to one. It seems that to calculate the factorial of a number is very easy – just multiply all the natural numbers not exceeding a certain. However, the value of the factorial grows so quickly that some calculators don't cope with this task.

You will need

- calculator, computer

Instruction

1

To calculate the factorial of a natural number, multiply all the natural numbers not exceeding this. Each number is counted only once. As a formula this can be written as follows:n! = 1*2*3*4*5*...*(n-2)*(n-1)*n, Gda is a natural number a factorial of which you want to calculate.

0! considered as equal to one (0!=1).With an increase of the argument the value of factorial increases very quickly, so the usual (financial) calculator for the factorial of 15 is it may give an error message.

0! considered as equal to one (0!=1).With an increase of the argument the value of factorial increases very quickly, so the usual (financial) calculator for the factorial of 15 is it may give an error message.

2

To calculate the factorial of large positive numbers, take a calculator. That is, the calculator on the keyboard which are marked with math functions (cos, sin, √). Enter to calculator original number, and then click calculate the factorial. Usually, this button is denoted as "n!" or similar (instead of the letter "n" might stand for "N" or "x", but the exclamation mark "!" factorial symbol should be present in any case).

For large values of the argument calculation results are displayed in "exponential" (exponential) form. For example, the factorial of 50 is presented in the form: e 3,0414093201713378043612608166065+64 (or similar). To get the result of the calculation in the normal manner, attribute to the number displayed before the "e" as many zeros as indicated after the "e+" (of course, if enough space).

For large values of the argument calculation results are displayed in "exponential" (exponential) form. For example, the factorial of 50 is presented in the form: e 3,0414093201713378043612608166065+64 (or similar). To get the result of the calculation in the normal manner, attribute to the number displayed before the "e" as many zeros as indicated after the "e+" (of course, if enough space).

3

To calculate the factorial of a number in the computer, run the calculator (standard Windows calculator). To do this, locate its image on the desktop or click on the button "start" and "Run". Then, type in the appeared window "calc" and hit "OK". View: how did you start the program "Calculator". If the picture resembles the ordinary "accounting" calculator, switch it to "engineering" mode. To do this, simply click on the item "View" and select in the list of options the string "Engineering".

After this, repeat the same steps that are listed in the previous paragraph of the instruction - enter the number and press "n!".

After this, repeat the same steps that are listed in the previous paragraph of the instruction - enter the number and press "n!".

4

"Calculate the factorial of a number is possible without the use of computer technology. To do this, simply print a table of factorials. Since the values of the factorial are increasing very rapidly, it is feasible to only print out factorials of numbers from 0 to 50. However, the practical application of such tables is very doubtful. Firstly, the entry of such multi-digit numbers will take a very long time, and secondly, the probability of entry errors, and, thirdly, it is not clear where to enter such a long number. Neither the display of the calculator or the Excel cell just barely fits so many figures.