Factorials
Factorials have been discovered in several ancient cultures, notably in Indian mathematics the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences - the permutations - of n distinct objects : there are n!, In mathematical analysis, factorials are under in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory and computer science.
A factorial is a function that multiples a number by every natural number below it till 1.
A factorial can be represented by using the symbol “!”.
“n factorial” is the product of the first n natural numbers and is represented as n! ( n factorial) n! = 1 x 2 x 3 x -------x n
= product of the first ‘n’ positive integers
= n (n-1) (n - 2) -----(3) (2) (1)
Ex : 4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
= 5 x 4! = 5 x 24 = 120
\(
\therefore
\) n! = n x ( n - 1) !
and also ( n +1)! = ( n +1 ) x n!
0! = 1 according the convention for an empty product
Empty product means the result of multiply no factors.
It is by convention equal to the multiplicative identity.
Number of trailing zeros in n! =
= no. of times n! is divided by 10
= Highest power of 10 which divides n!
= no. of 5’s in n!
\(
= \left( {\frac{n}
{5}} \right) + \left( {\frac{n}
{{25}}} \right) + \left( {\frac{n}
{{125}}} \right) + ........
\)
In doing this we consider the quotient as whole number if we get a decimal.
Ex : [4.01] = 4, [0.1] = 0
We use another simple method to count no. of trailing zeros i.e.,
Divide the number ‘n’ in n! by 5 and its subsequent quotients by 5 as long as the quotient is non-zero (each time ignore any non-zero remainder). Finally, add up all these non zero quotients and that will be the no. o5’s in n!
Ex : No. of trailing zeros at the end of 100!.
\(
\frac{{100}}
{5} = 20
\)
\(
\frac{{20}}
{5} = 4
\) 4 < 5 stop here
\(
\therefore
\) No. of trailing zeros are 20 + 4= 24