When a player buys a Mega Millions ticket in many states, the player can also buy the Megaplier, which multiplies the size of a prize other than a jackpot by a multiplier ranging from two to five. The Megaplier is drawn using a pool of 15 balls, with five marked 2X, six marked 3X, three marked 4X, and one marked 5X, where each ball has the same likelihood of being drawn. Find the probability that a player who buys a Mega Millions ticket and the Megaplier wins $20? (The three ways to do this are to match exactly three of the first five numbers drawn, but not the sixth number drawn, or exactly two of the first five numbers and the sixth number, with Megaplier 2X, or to match exactly one of the first five numbers and the sixth number, with Megaplier 5X.)
Note that there are 5 balls marked with 2X and 1 ball marked with 5X out of total 15 Megaplier balls. $\def\c(#1,#2){\binom{#1}{#2}}$
My attmept: For the first way the probability is, $$\c(5,3)\c(65,2) 24 \over \c(70,5)\c(25,1)$$
For the second way the probability is, $$\c(5,2)\c(65,3) 5 \cdot 1 \over \c(70,5)\c(25,1)\c(15,1)$$
For the third way the probability is, $$\c(5,1)\c(65,4) 1 \cdot 1 \over \c(70,5)\c(25,1)\c(15,1)$$
As these events are disjoint(the intersections of the events are null), the probability of the union of these three events are found by adding the individual probabilities,
$$\frac{499,200}{302,575,350} + \frac{2,184,000}{4,538,630,250}+\frac{3,385,200}{4,538,630,250}=\frac{24,648}{10,085,845}$$
The indicated answer in the book by K. Rosen is $\frac{4888}{2,750,685}$. Where have I gone wrong?
EDIT: As answered by Pavan, ambiguity of english and an oversight in the answer by me(without the 2X megaplier, you win 10 dollars instead) caused the error. The first probability, when considering the 2X megaplier gives, $$ \c(5,3)\c(65,2) 24 \cdot 5 \over \c(70,5)\c(25,1) \c(15,1)$$ therefore the new sum is $$\frac{2,496,000}{4,538,630,250} + \frac{2,184,000}{4,538,630,250}+\frac{3,385,200}{4,538,630,250}=\frac{4888}{2,750,685}$$
To clear a few semantics: I think you used the word independent at the end when you meant disjoint. Also other than that I think you made a small math error in the second numerator: it should be $2,\!184,\!000$.
Unfortunately, the words in the text are unclear and I can see why you did what you did. In other words, you did the problem absolutely right, so this is not a problem in understanding--it's a problem in English!
The error would be clear if you checked the MegaMillions site. If you check it, you would see that you actually have these three options:
You missed the bolded part in the first line, due to the confusing writing in the question. Apply your same logic and you will get the right answer. Hope this helps!