Combination Beads There are $4$ beads: white, blue, green, red. I can have at most $8$ beads. Each sequence of up to $8$ beads indicate a different message. It can have repeated elements.
A) What are the number of distinct messages? 
B) If I buy $10$ beads, how many ways are there to purchase different beeds?
I am confused how I would go about solving this problem. I got it as $\binom{8}{4}$ but it seems wrong.

A) We include the empty message. 
A message is a string (word) made up of letters from the $4$-letter alphabet {W,B,G,R}.  
The number of messages of length $0$ is $1$, the number of messages (words) of length $1$ is $4$, the number of messages of length $2$ is $4^2$, and so on up to $4^8$. The total is therefore $1+4^1+4^2+\cdots+4^8$. Using the expression for the sum of a finite geometric series, the number can be written as $\frac{4^{9}-1}{3}$. If we wish not to allow the empty message, the number is $\frac{4^{9}-4}{3}$.
B) This is a problem quickly solved using the Stars and Bars method.  We want to find the number of solutions of the equation $x_1+x_2+x_3+x_4=10$ in non-negative integers. The Wikipedia article linked to is quite good.   
