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April 9th, 2014, 07:00 PM  #1 
Senior Member Joined: Jan 2009 Posts: 345 Thanks: 3  How many sequences contain 2 Tails and 3 Heads
Confused about this one. I flip a coin 5 times. How many sequences have 2 tails and 3 heads? Is it Binomial[5,2]+Binomial[5,3] = 20? 
April 9th, 2014, 11:32 PM  #2 
Senior Member Joined: Apr 2014 From: UK Posts: 921 Thanks: 331 
I used fingermaths and added them up manually. these are the sequences with 2 tails 1,2 1,3 1,4 1,5 2,3 2,4 2,5 3,4 3,5 4,5 So the answer is 10 This is probably your Binomial[5,2]. The other half isn't required as it's implied, with 2 tails the other 3 must be heads anyway 
April 10th, 2014, 12:04 AM  #3 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
As there are only a handful of possibilities, we can list them all exhaustively: $\text{TTHHHH}$So there are $10$ possibilities. In general if a coin is thrown $n$ times, any sequence with exactly $r$ tails with have exactly $nr$ heads and vice versa. So to find the number of sequences with exactly $r$ tails, compute EITHER $^n\mathrm C_r$ (tails) OR $^n\mathrm C_{nr}$ (heads). 

Tags 
binomial, heads, sequences, tails 
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