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MMM #23: Counting coin runs

Yes, I like counting problems. I have a Rubik’s Revolution, courtesy of Techno Source (or $10 Amazon.com gift certificate, if you prefer, or $10 in USD via PayPal to non-US folks) to give to the winner.

Here's the problem:

If you toss a fair coin 20 times, in how many ways can you get a run of exactly 10 heads? In other words, in how many ways can exactly 10 coins in 10 consecutive tosses to land heads side up out of a total of 20 tosses?

This is a valid way that you can get a run of 10 heads in 20 tosses:
   HHHHHHTHHHHHHHHHHTHH (there are 18 heads, but exactly 10 in a row)

This is an INVALID result:
   HHHHHHHHHTHTTTTTTTTT (there are 10 heads, but not all are consecutive)

This is an INVALID result:
   HHHHHHHHHHHTTHHTHTHH (there are 11 consecutive heads)

Here are the rules for the contest:

1. Email your answers with solutions to mondaymathmadness at gmail dot com.
2. Only one entry per person.
3. Each person may only win one prize per 12 month period. But, do submit your solutions even if you are not eligible.
4. Your answer must be explained. You must show your work! Wild About Math! and Blinkdagger will be the final judges on whether an answer was properly explained or not.
5. The deadline to submit answers is Tuesday, January 13, 12:01AM, Pacific Time. (That’s Tuesday morning, not Tuesday night.) Do a Google search for “time California” to know what the current Pacific Time is.)
6. The winner will be chosen randomly from all timely well-explained and correct submissions, using a random number generator.
7. The winner will be announced Friday, January 16, 2009.
8. The winner (or winners) will receive a Rubik’s Revolution or a $10 gift certificate to Amazon.com or $10 USD via PayPal. For those of you who don’t want a prize I’ll donate $10 to your favorite charity.
9. Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.
10. I may post names and website/blog links for people submitting timely correct well-explained solutions. I’m more likely to post your name if your solution is unique.

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  1. Just to clarify, when you say “10 consecutive tosses – no more and no less – land heads side up” do you mean that only the coins in the run of 10 are heads (i.e. all the rest are tails), or that the run has exactly 10 heads (i.e. there cannot be a head adjacent to the run of 10 heads)?

  2. John, I mean the latter. Thanks for the clarification. I’ve updated the problem description.

    If anyone got the wrong answer because of my lack of clarity, I’ll send them an email and give them another chance to solve it.

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