Binomial Theorem

• Jan 5th 2012, 11:00 AM
kangta27
Binomial Theorem
In the expansion of (x+y)^n the coefficient of x^4y^(n-4) is 3,876 and the coefficient of x^5y^(n-5) is 11,628. Find the coefficient of x^5y^(n-4) in the expansion of (x+y)^(n+1). What is the value of n?
• Jan 5th 2012, 03:20 PM
Ridley
Re: Binomial Theorem
Each coefficient can be calculated as follows: $\displaystyle a_i=\frac{n!}{i!(n-i)!}$

$\displaystyle a_4=\frac{n!}{4!(n-4)!}=3876 \Leftrightarrow n! = 3876 \cdot 4! \cdot (n-4)!$ (1)
$\displaystyle a_5=\frac{n!}{5!(n-5)!}=11628 \Leftrightarrow n! = 11628 \cdot 5! \cdot (n-5)!$ (2)

Combine (1) and (2):
$\displaystyle 3876 \cdot 4! \cdot (n-4)! = 11628 \cdot 5! \cdot (n-5)! \Leftrightarrow n-4 = \frac{11628 \cdot 5!}{3876 \cdot 4!}$
• Jan 5th 2012, 03:39 PM
kangta27
Re: Binomial Theorem
Thank you so much Ridley! My only question is, what happened to the (n-4)! and (n-5)! when you combined (1) and (2)?
• Jan 5th 2012, 03:44 PM
Ridley
Re: Binomial Theorem
$\displaystyle (n-4)!= (n-4)\cdot(n-5)\cdot(n-6)\cdot...\cdot1 = (n-4) \cdot (n-5)!$