Just a query re: simplifying n!/[(k - 1)!(n + 1 - k)!]

[morson]

Is there a means of simplifying the expression n!/[(k - 1)!(n + 1 - k)!] ?

I have been told to find a general expansion of (a + b)^n, and I have found the expression: n!/[(k - 1)!(n + 1 - k)!] * a^{n+1-k} * b^{k-1} to be the k'th term.

 

[stapel]

I have been told to find a general expansion of (a + b)^n....

This is just my guess, but I would suspect that the form you're using is as "simplified" as they're expecting.

I could be wrong, of course...

Eliz.

 

[JakeD]

Is there a means of simplifying the expression n!/[(k - 1)!(n + 1 - k)!] ?

I have been told to find a general expansion of (a + b)^n, and I have found the expression: n!/[(k - 1)!(n + 1 - k)!] * a^{n+1-k} * b^{k-1} to be the k'th term.

\(\displaystyle (a+b)^n\) is a binomial expansion and \(\displaystyle \frac{n!}{(k - 1)!(n + 1 - k)!}\) is a binomial coefficient, denoted \(\displaystyle {n \choose k-1}.\)

 

[morson]

\(\displaystyle (a+b)^n\) is a binomial expansion and \(\displaystyle \frac{n!}{(k - 1)!(n + 1 - k)!}\) is a binomial coefficient, denoted \(\displaystyle {n \choose k-1}.\)

Brilliant, thanks.