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 sivela April 9th, 2014 07:00 PM

How many sequences contain 2 Tails and 3 Heads

I flip a coin 5 times. How many sequences have 2 tails and 3 heads?
Is it Binomial[5,2]+Binomial[5,3] = 20?

 weirddave April 9th, 2014 11:32 PM

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
The other half isn't required as it's implied, with 2 tails the other 3 must be heads anyway

 Olinguito April 10th, 2014 12:04 AM

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).

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