Permutation and Combination Question?

let me solve in a very simple way
transform the problem to putting 10 identical balls into 4 coloured boxes, then by the "stars and bars" formula (see link), we directly get the answer
# of ways = (10+4-1)C(4-1) = 13C3 (or 13C10) = 286 <------- qed

http://jhyun95.hubpages.com/hub/Stars-and-Bars-Combinatorics

Number of ways to select 10 balls from n balls = nP10 = n!/(n-10)!
but you are given infinite number of balls.
how it is possible to choose
It cannot be infinite.

Your book used a generating function.
x is a dummy variable, the x terms are just placeholders for the coefficients and the coefficients are the numbers you're interested in.
There are 2 free electronic books on generating functions (you can google them).
"Generatingfunctionology" by Wilf
"Analytic Combinatorics" by Flajolet

And if your textbook used a generating function for a solution, chances are it explains them in one of the chapters. Read that first, then if you're not satisfied, read some of Wilf (which is probably easier to read than Flajolet).

A generating function is just one way to solve the problem. These kinds of problems are known as multiset combinations. The formula is C(n+r-1, r) where n is the number of possible choices (in this case 4) and r is the number of items being selected (10). Wikipedia has a good combinatorial explanation of why the formula works (where it uses dots and bars) -- https://en.wikipedia.org/wiki/Multiset#Counting_multisets
Personally I prefer that approach for this problem, though generating functions are also nice and can certainly come in handy.

The question is here :

http://books.google.co.in/books?id=u7HWgE60km4C&pg=SA7-PA7&lpg=SA7-PA7&dq=The+number+of+ways+of+choosing+10+balls+from+infinite+white,+red,+blue,+green+balls+is+?&source=bl&ots=re_xsabohq&sig=tLa8g_dwlf0DtiyKUDAqmpK4nEw&hl=en&sa=X&ei=kMDXUt2-EYWFrgf3ooHYCg&ved=0CCoQ6AEwAA#v=onepage&q=The%20number%20of%20ways%20of%20choosing%2010%20balls%20from%20infinite%20white%2C%20red%2C%20blue%2C%20green%20balls%20is%20%3F&f=false

It should be
4 * 4 * ... * 4 (ten 4s) = 4^10.