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Factorials
Factorials
1. Factorial Notation : The product of first n natural numbers is denoted by and is read as “factorial n”.This
\( n! = 1 \times 2 \times 3 \times 4 \times ............ \times \left( {n - 1} \right) \times n \)
\( = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ......... \times 3 \times 2 \times 1. \)
For example, \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \)
\( = 120 \)
\( 4! = 1 \times 2 \times 3 \times 4 \)
\( = 24 \)
2. The Factorial Of Zero: The factorial of 0 i.e \( 0! \) is defined as 1. i.e. \( 0! = 1 \)
3. The following values should be remembered.
Note : (1) Except \( 0! \), \( 1! \), the factorial of every natural number is an even number.
(2) From onwards, the units digit is zero.
4. Some Results Related To Factorial n :
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Factorials
Factorials
1. Factorial Notation : The product of first n natural numbers is denoted by and is read as “factorial n”.This
\( n! = 1 \times 2 \times 3 \times 4 \times ............ \times \left( {n - 1} \right) \times n \)
\( = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ......... \times 3 \times 2 \times 1. \)
For example, \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \)
\( = 120 \)
\( 4! = 1 \times 2 \times 3 \times 4 \)
\( = 24 \)
2. The Factorial Of Zero: The factorial of 0 i.e \( 0! \) is defined as 1. i.e. \( 0! = 1 \)
3. The following values should be remembered.
Note : (1) Except \( 0! \), \( 1! \), the factorial of every natural number is an even number.
(2) From onwards, the units digit is zero.
4. Some Results Related To Factorial n :
(i) \( n! = 1 \times 2 \times 3 \times 4 \times ............ \times \left( {n - 1} \right) \times n \)
\( = \left[ {1 \times 2 \times 3 \times ........ \times \left( {n - 1} \right)} \right] \times n \)
\( = \left( {n - 1} \right)! \times n \)
Hence, \( n! = \left( {n - 1} \right)! \times n \)
\( = n \times \left( {n - 1} \right)! \)
Similarly
Thus \( n! = n\left( {n - 1} \right)! \)
\( = n\left( {n - 1} \right)\left( {n - 2} \right)! \)
\( = n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)! \) and so on.
e.g.(i) \( 10! = 10 \times 9! \)
\( = 10 \times 9 \times 8! \)
\( = 10 \times 9 \times 8 \times 7! \) and so on.
(ii) If n and r two positive integers such that \( r < n, \) then
\( \frac{{n!}} {{r!}} = \frac{{1 \times 2 \times 3 \times ......r \times \left( {r + 1} \right)\left( {r + 2} \right).... \times n}} {{1 \times 2 \times 3 \times ..... \times \left( {r - 1} \right) \times r}} \)
\( = \left( {r + 1} \right)\left( {r + 2} \right).........\left( {n - 1} \right) \times n \)
\( = n\left( {n - 1} \right)\left( {n - 2} \right)......\left( {r + 1} \right) \)
(iii) \( \frac{n} {{\left( {n - r} \right)!}} = \frac{{1 \times 2 \times 3 \times ... \times \left( {n - 1} \right) \times n}} {{1 \times 2 \times 3 \times ... \times \left( {n - r} \right)}} \)
\( = \frac{{1 \times 2 \times 3 \times ... \times \left( {n - r} \right)\left( {n - r + 1} \right)\left( {n - r + 2} \right) \times ... \times \left( {n - 1} \right)n}} {{1 \times 2 \times 3 \times ..... \times \left( {n - r} \right)}}\)
\( = \left( {n - r + 1} \right)\left( {n - r + 2} \right)....\left( {n - r} \right)n \)
\( = n\left( {n - 1} \right)\left( {n - 2} \right).....\left( {n - r + 2} \right)\left( {n - r + 1} \right) \)
\( = n\left( {n - 1} \right)\left( {n - 2} \right)...... \) to r factors
(iv) \( (2n)! = 2^n n!\left( {1.3.5.7....\left( {2n - 1} \right)} \right) \)