Factorial Notation, Illustration, Rapid Questions, Examples, Important Topics For JEE Main 2024
Shrivastav 8 Feb, 2024
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Factorial Notation : Factorial is originated from the French word factorielle. At ancient time many countries made use of it in various calculations such as in Indian mathematics it was used between 300BCE to 400CE in Jain literature later in 6 th -century CE Jinabhadra described product rule of factorization, Bhaskara II used factorial in his work Lilavati. Arab Mathematician Alhazen formulate Wilson’s theorem having use of factorial with prime numbers.in 1677 British author Fabian Stedman made use of factorial in musical art known as change ringing.
Factorial is defined for whole numbers and represents product of whole numbers staring from 1 as factorial of 0 is defined as 1 it helps in representing product of n natural numbers in short form such as n! could be written as
, so value of 6! by the same method would be 6
=720 , real life use of factorial is in the implementation of fundamental rule of multiplication which described as if any task is divided in subtasks and each subtask is independent of each other than all individual outcomes must be multiplied to obtain the final result.
For example. if a person has to make word using 4 alphabets A, B, C, D (each one can be used once) than total number of words would be
or
,Let’s explore factorial notation with the help of examples and illustrations in next section.
Factorial Notation : As we have discussed continuous product of first n natural numbers is known as factorial value of n! is even except the case when n is 1, factorial is undefined for fractions or negative numbers, value of 0! is one as n! leads to consecutive product of positive integers with difference of one so it will lead to 0 and multiplication of 0 with any value made the result as 0 which does not make sense in factorial hence value of 0! Is fixed as 1.
Factorial Notation Examples
let’s see some examples.Example 1: Find LCM of 7!, 3!, 5!?
Solution: Terms LCM is defined as least common product means a value which is first multiple of all the terms combined for example LCM of 4, 6 is 12 which is combinedly divisible by 4, 6.
Now 7! =
,
,
LCM would be
(3! Common in all, next common in two is 5
and remaining is
)
Example 2:
Find value of x if
Solution:
Above equation could be written as
Example 3:
Find value of x if
Solution: Using factorial expansion Above term could be written as
By comparing both sides value of x would be 40.
Example 4:
Convert the given product in factorial form
Solution:
Factorial is defined as continuous product of first n natural numbers
(
is multiplied in both numerator and denominator)
Example 5:
Convert the given product in factorial form
Solution:
Factorial is defined as continuous product of first n natural numbers
(
Factorial Notation Rapid Question
(1) Convert the given product in factorial form
?
(2) Find value of x if
?
(3) Find value of x if
?
Factorial Illustration
(1) Find the value of x if
Solution:
By comparing both sides value of x would be 6.
(1) If ration of
and
is
find
Solution:
=
could be written as
Now ration of both would be
(2) If ration of
and
is
find
Solution:
Now ration of both would be
by comparing both sides x =7
(3) Prove that ration of
and
is
Solution:
Factorial Illustration And Examples
Factorial Illustration : Factorial is originated from the French word factorielle. At ancient time many countries made use of it in various calculations such as in Indian mathematics it was used between 300BCE to 400CE in Jain literature later in 6 th -century CE Jinabhadra described product rule of factorization, Bhaskara II used factorial in his work Lilavati. Arab Mathematician Alhazen formulate Wilson’s theorem having use of factorial with prime numbers.in 1677 British author Fabian Stedman made use of factorial in musical art known as change ringing .
Factorial is defined for whole numbers and represents product of whole numbers staring from 1 as factorial of 0 is defined as 1, it helps in representing product of n natural numbers in short form such as n! and could be written as
, so value of 4! by the same method would be
=24 , real life use of factorial is in the implementation of fundamental rule of multiplication which described as if any task is divided in subtasks and each subtask is independent of each other than all individual outcomes must be multiplied to obtain the final result.
For example, if a person has to make word using 3 alphabets A, B, C (each one can be used once) than total number of words would be
or
,Let’s explore factorial notation with the help of examples and illustrations in next section.
Factorial Illustration Introduction
Factorial Illustration Introduction: As we have discussed continuous product of first n natural numbers is known as factorial, Factorial has some standard results which need to be remember.
-
Factorial does not follow distributive law when terms attached with addition, subtraction, multiplication and division symbols
and
-
Product of n consecutive numbers is always divisible by n! for example
has 4 terms than it must be divisible by
- (1! +2! +3! +………. +n!) is perfect square only when n=1,3 this is because
Factorial Illustration Examples
let’s see some examples.
Example 1:
Find the value of (3+4)! And prove that it is not equal to
Solution:
Value of
Value of 4! is
Sum of 3! and 4! is 30 while
hence proof.
Example 2:
Find the value of (
Solution-
Value of
Value of 2! is
Product of 3! and 2! is 12 while
hence proof.
Example 3:
Find the value of (
Solution:
Value of (
Value of 2! is
Division of 6! and 2! is 360 while
hence proof.
Example 4.
Convert the given product in factorial form
Solution:
Factorial is defined as continuous product of first n natural numbers
(
is multiplied in both numerator and denominator)
Example 5:
Convert the given product in factorial form
Solution:
Factorial is defined as continuous product of first n natural numbers
Factorial Illustration Rapid Question
(1) Convert the given product in factorial form
?
(2) Find value of x if
?
(3) Find value of x if
?
Illustration:
(1) Find the value of x if
Solution:
By comparing both sides value of x would be 6.
(2) If ration of
and
is
find
Solution:
=
could be written as
Now ration of both would be
(3) If ration of
and
is
find
Solution:
Now ration of both would be
by comparing both sides x =7
(4) Prove that ration of
and
is
Solution:
Q.1: Factorial is originated from the Greek word factorielle.?
Q. 2: Bhaskara II made use of factorial in his work Lilavati?
Q.3: Factorial is defined as continuous product of natural numbers?
Q. 4. Factorial of fractions exist and equal to product of consecutive fractions with difference of 1?
Q.5. Factorial always gives even value?
Q.6: Factorial is originated from the Greek word factorielle.?
Q.7: Bhaskara II made use of factorial in his work Lilavati?
Q.8. Factorial is defined as continuous product of natural numbers?
Q. 9. Factorial of fractions exist and equal to product of consecutive fractions with difference of 1?
Q.10. Factorial always gives even value?
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