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23

FACTORIALS

BY THE SYMBOL n! ("n factorial") we mean the product of consecutive numbers 1 through n.

n!  =  1· 2· 3·  ·  · (n − 2)(n − 1)n
 
   =  n(n − 1)(n − 2)·  ·  · 3· 2· 1

The order of the factors does not matter, whether backwards or forwards.

0! is defined as 1.

0! = 1

(The usefulness of this definition will become clear as we continue.)

Example.   6! = 1· 2· 3· 4· 5· 6 = 720

A convenient way to calculate this is to wait to multiply 2· 5 = 10.  Then

3· 4· 6 = 12· 6 = 72

-- times 10 is 720.

Calculations with factorials are based on this fact:

Any factorial less than n! is a factor of n!.

Example 1.   6! is a factor of 10!.  For,

10!  =  1· 2· 3· 4· 5· 6· 7· 8· 9· 10
 
   =  6!· 7· 8· 9· 10
   Example 2.   Evaluate   8!
5!
.
8!
5!
  =   1· 2· 3· 4· 5· 6· 7· 8
      1· 2· 3· 4· 5
  =   6· 7· 8   =   6· 56   =  336
   Example 3.   Evaluate     8!  
6! 2!

Solution.  6! is a factor of 8!, so it will cancel  leaving 7· 8 in the

  numerator.  We will have   7· 8
1· 2
  =  28.

Problem 1.   What factorial is each of these?  (Consider what each symbol means.)

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a)  3!· 4  = 4!  For, 3!· 4 = 1· 2· 3· 4 = 4!.         b)  6!· 7· 8  = 8! 

c)  (n − 1)! n  = n!              d)  (nk − 1)! (nk) = (nk)!

   Problem 2.   Evaluate   10!
 7!
.
10!
 7!
  =   7!· 8· 9· 10
      7!
  =   8· 9· 10   =   720
   Problem 3.   Evaluate      12!  
 10! 2!
.
   12!  
 10! 2!
  =   10!· 11· 12
   10! 2!
  =   11· 12
  1· 2
  =   11· 6 = 66
   Example 4.   Show:       1     
(n − 1)!
  =   n 
n!

Solution.  If we multiply both terms on the left by n, then

     1     
(n − 1)!
  =         n      
(n − 1)! n

Now, (n − 1)! is the product up to the number just before n.  Therefore, (n − 1)! times n itself  is n!.

     1     
(n − 1)!
  =         n      
(n − 1)! n
  =   n 
n!

Example 5.   Show:  n! k  +  n! (nk + 1) = (n + 1)!

 Solution.   On the left-hand side, n! is a common factor:

n! k  +  n! (nk + 1)  =  n! (k + nk + 1)
 
   =  n! (n + 1),   on canceling the k's,
 
   =  (n + 1)!

For, (n + 1) is the number after n.

Problem 4.   Write out the steps that show how to transform the left-hand side into the right-hand side.  Again, consider what each symbol means.

  a)     1
3!
  =   4
4!

Multiply both terms on the left by 4:

 1
3!
  =      4  
3!· 4
  =    4
4!
  b)         1     
(k − 1)!
  =   k 
k!

Multiply both terms on the left by k:

     1    
(k − 1)!
  =          k      
(k − 1)!· k
  =   k 
k!
  c)    12 
12!
  =    1 
11!

Since 12! = 11!· 12, divide both terms on the left by 12:

12 
12!
  =      12   
11!· 12
  =    1 
11!
  d)     nk  
(nk)!
  =           1        
(nk − 1)!

Since (nk)! = (nk − 1)!· (nk), divide both terms on the left by (nk):

 nk  
(nk)!
  =             nk           
(nk − 1)!· (nk)
  =           1        
(nk − 1)!

e)   5!· 4  +  5!· 2 = 6!

5! is a common factor:

5!· 4  +  5!· 2 = 5!(4 + 2) = 5!· 6 = 6!


Next Topic:  Permutations and combinations


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