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Binomial coefficient

  • #1

Homework Statement



How is possible this equality:

[tex] {n \choose k} = \frac{n \cdot (n-1) \cdots (n-k+1)}{k(k-1)...1} = \frac{n!}{k!(n-k)!}[/tex]

? I mean where the second part [tex]\frac{n!}{k!(n-k)!}[/tex] comes from?

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
761
13
[tex] n! = n \cdot (n-1) \cdot (n-2) \cdot \cdot \cdot (n-k-1) \cdot (n-k-2) \cdot \cdot \cdot 3 \cdot 2 \cdot 1 = n \cdot (n-1) \cdot (n-2) \cdot \cdot \cdot (n-(k-1)) \cdot (n-k)![/tex]

[tex] k! = k \cdot (k-1) \cdot (k-2) \cdot \cdot \cdot 2 \cdot 1 [/tex]


So we get:

[tex] \frac{n!}{(n-k)!} = n \cdot (n-1) \cdot \cdot \cdot n-k+1 [/tex]
 
  • #3
210
0
multiply the first bit by (n-k)!/(n-k)! :smile:
 
Last edited:
  • #4
Ok, I understand about:

[tex]
\frac{n!}{(n-k)!} = n \cdot (n-1) \cdot \cdot \cdot n-k+1
[/tex]
But still I can't understand why n! is written like that..
 
  • #5
761
13
Ok, I understand about:

[tex]
\frac{n!}{(n-k)!} = n \cdot (n-1) \cdot \cdot \cdot n-k+1
[/tex]
But still I can't understand why n! is written like that..
Note that:

[tex] {n \choose k} \neq \frac{n!}{k!} [/tex]

The definition is the one in your first post.
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,795
925
Ok, I understand about:

[tex]
\frac{n!}{(n-k)!} = n \cdot (n-1) \cdot \cdot \cdot n-k+1
[/tex]
But still I can't understand why n! is written like that..
Written like what?

You started by saying that you understand that
[tex] {n \choose k} = \frac{n \cdot (n-1) \cdots (n-k+1)}{k(k-1)...1}[/tex]
but did not understand why
[tex]\frac{n \cdot (n-1) \cdots (n-k+1)}{k(k-1)...1}= \frac{n!}{k!(n-k)!}[/tex]
That is what was just explained. It is written "that way" in order to give a simple closed form expression to
[tex]\frac{n \cdot (n-1) \cdots (n-k+1)}{k(k-1)...1}[/tex]
"without the dots".
 

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