Just Throwing it Out There

I have a statistics problem that I don’t know if I am solving correctly.  I am asking the internet to either verify my work or correct it. 

Here’s the problem:

What are the odds of dealing a specific sequence of 18 cards from a standard, randomized 52 card deck?  Stated another way, there are 9 poker players sitting around a table.  In the game of Texas Hold’em each player is dealt 2 cards.  The dealer deals each player one card and then starts over again dealing the second.  What is the probability a specific order of cards (both in value and in suit) is dealt?  Of course the situation doesn’t matter since it is really just the first 18 cards out of the deck but I thought I would clarify the question.  The reason why I want to know this will be discussed AFTER the answer is determined.

Here is my solution:

The odds of any one card coming out 1st is 1 out of 52.  The odds for the 2nd card are 1 out of 51.  This goes on until all 18 cards are dealt.  So, the math is simply (in my thinking at least) is 52 * 51 * 50 * 49…all the way to 35.  This gives an answer of 2.73 x 10^29 to 1 against.

Am I close?

And just FYI…there reason I want to know the answer has nothing to do with poker or gambling.  I am just using the card example to help make a point about something entirely unrelated.

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