SOLUTION: How many simple events does the sample space contain when n coins are thrown m times?

Here is the sample space when 2 coins are thrown 3 times:

{(HH,HH,HH), (HH,HH,HT), (HH,HH,TH), (HH,HH,TT),
 (HH,HT,HH), (HH,HT,HT), (HH,HT,TH), (HH,HT,TT),
 (HH,TH,HH), (HH,TH,HT), (HH,TH,TH), (HH,TH,TT),
 (HH,TT,HH), (HH,TT,HT), (HH,TT,TH), (HH,TT,TT),
 (HT,HH,HH), (HT,HH,HT), (HT,HH,TH), (HT,HH,TT),
 (HT,HT,HH), (HT,HT,HT), (HT,HT,TH), (HT,HT,TT),
 (HT,TH,HH), (HT,TH,HT), (HT,TH,TH), (HT,TH,TT),
 (HT,TT,HH), (HT,TT,HT), (HT,TT,TH), (HT,TT,TT),
 (TH,HH,HH), (TH,HH,HT), (TH,HH,TH), (TH,HH,TT),
 (TH,HT,HH), (TH,HT,HT), (TH,HT,TH), (TH,HT,TT),
 (TH,TH,HH), (TH,TH,HT), (TH,TH,TH), (TH,TH,TT),
 (TH,TT,HH), (TH,TT,HT), (TH,TT,TH), (TH,TT,TT),
 (TT,HH,HH), (TT,HH,HT), (TT,HH,TH), (TT,HH,TT),
 (TT,HT,HH), (TT,HT,HT), (TT,HT,TH), (TT,HT,TT),
 (TT,TH,HH), (TT,TH,HT), (TT,TH,TH), (TT,TH,TT),
 (TT,TT,HH), (TT,TT,HT), (TT,TT,TH), (TT,TT,TT))

Each simple event consists of a group of 3 tosses each, and
each contains 2 coin tosses, so there are 
2×3 or 6 places to put H or T. So there are 22×3
elements in the sample space. 22×3 = 26 = 64.

Here is the sample space when 3 coins are thrown 2 times:

{(HHH,HHH),  (HHH,HHT),  (HHH,HTH),  (HHH,HTT),
 (HHH,THH),  (HHH,THT),  (HHH,TTH),  (HHH,TTT),
 (HHT,HHH),  (HHT,HHT),  (HHT,HTH),  (HHT,HTT),
 (HHT,THH),  (HHT,THT),  (HHT,TTH),  (HHT,TTT),
 (HTH,HHH),  (HTH,HHT),  (HTH,HTH),  (HTH,HTT),
 (HTH,THH),  (HTH,THT),  (HTH,TTH),  (HTH,TTT),
 (HTT,HHH),  (HTT,HHT),  (HTT,HTH),  (HTT,HTT),
 (HTT,THH),  (HTT,THT),  (HTT,TTH),  (HTT,TTT),
 (THH,HHH),  (THH,HHT),  (THH,HTH),  (THH,HTT),
 (THH,THH),  (THH,THT),  (THH,TTH),  (THH,TTT),
 (THT,HHH),  (THT,HHT),  (THT,HTH),  (THT,HTT),
 (THT,THH),  (THT,THT),  (THT,TTH),  (THT,TTT),
 (TTH,HHH),  (TTH,HHT),  (TTH,HTH),  (TTH,HTT),
 (TTH,THH),  (TTH,THT),  (TTH,TTH),  (TTH,TTT),
 (TTT,HHH),  (TTT,HHT),  (TTT,HTH),  (TTT,HTT),
 (TTT,THH),  (TTT,THT),  (TTT,TTH),  (TTT,TTT)}


Each simple event consists of a group of 2 tosses each, and
each contains 3 coin tosses, so there are 
3×2 or 6 places to put H or T. So there are 23×2
elements in the sample space. 22×3 = 26 = 64.

Now if we were to list the sample space when n coins are thrown m times,
                                             
Each simple event consists of a group of m tosses each, and
each contains n choices of coin tosses, so there are 
n×m places to put H or T. So there are 2n×m
elements in the sample space.

Edwin