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?
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):
$\displaystyle 3876 \cdot 4! \cdot (n-4)! = 11628 \cdot 5! \cdot (n-5)! \Leftrightarrow n-4 = \frac{11628 \cdot 5!}{3876 \cdot 4!}$