My Math Forum How many sequences contain 2 Tails and 3 Heads

 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: 965 Thanks: 342 I used finger-maths 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: $\text{TTHHHH}$ $\text{THTHHH}$ $\text{HTTHHH}$ $\text{THHTHH}$ $\text{HTHTHH}$ $\text{HHTTHH}$ $\text{THHHTH}$ $\text{HTHHTH}$ $\text{HHTHTH}$ $\text{HHHTTH}$ $\text{THHHHT}$ $\text{HTHHHT}$ $\text{HHTHHT}$ $\text{HHHTHT}$ $\text{HHHHTT}$So there are $10$ possibilities. In general if a coin is thrown $n$ times, any sequence with exactly $r$ tails with have exactly $n-r$ heads and vice versa. So to find the number of sequences with exactly $r$ tails, compute EITHER $^n\mathrm C_r$ (tails) OR $^n\mathrm C_{n-r}$ (heads). Thanks from sivela

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