A good way to visualize all the possibilities is to think of the possibilities in sets, where the run of at least five consecutive heads starts at some toss, n, and a set includes all possibilities for some n. The sets of permutations can be specified by placing the first head of the run of at least five consecutive heads at some place, n, right after a tail (excluding n=1), where the sets are ordered for n=1,2,3,4,5, and 6 in the following way:
H, H, H, H, H, ..., ..., ..., ..., ... respresents the set of all combinations whose runs of at least five consecutive heads start with the first toss. The following five tosses can occur in any way; consequently, there are 2x2x2x2x2 = 32 ways these can occur.
T, H, H, H, H, H, ..., ..., ..., ... represents the set of all combinations whose runs of at least five consecutive heads start with the second toss. The following four tosses can occur in any way; consequently, there are 2x2x2x2 = 16 ways these can occur.
..., T, H, H, H, H, H, ..., ..., ... represents the set of all combinations whose runs of at least five consecutive heads start with the third toss. The first toss can occur in two ways, and the last three can occur in any way; consequently, there are 2x2x2x2 = 16 ways these can occur.
..., ..., T, H, H, H, H, H, ..., ... represents the set of all combinations whose runs of at least five consecutive heads start with the fourth toss. The first two tosses can occur in any way, and the last two can occur in any way; consequently, there are 2x2x2x2 = 16 ways these can occur.
..., ..., ..., T, H, H, H, H, H, ... represents the set of all combinations whose runs of at least five consecutive heads start with the fifth toss. The first three tosses can occur in any way, and the last one can occur in two ways; consequently, there are 2x2x2x2 = 16 ways these can occur.
..., ..., ..., ..., T, H, H, H, H, H represents the set of all combinations whose runs of at least five consecutive heads start with the sixth toss. The first four tosses can occur in any way; consequently, there are 2x2x2x2 = 16 ways these can occur.
By adding 32 + 16 + 16 + 16 + 16 + 16, it then follows that there are 112 distinct combinations (permutations; order matters) of ten tosses, each with a run of at least five consecutive heads. Also, there are 2x2x2x2x2x2x2x2x2x2 = 1024 ways a coin can be tossed ten times. Therefore, the probability of getting a run of at least five consecutive heads in ten tosses of a coin is 112/1024 = .109375 or 10.9375 %. Keep in mind that probability is a fancy term for the long term relative frequency of an event of a random phenomenon and is what one would tend to observe in a very long series of trials. If anyone is so inclined to verify this, try tossing a coin one thousand times and recording the number of sets of ten tosses out of one hundred where there were runs of at least five consecutive heads. If possible, schedule a coin toss event and have a group of friends, classmates, etc. carry out sets of ten tosses simultaneously to expedite this otherwise slow process. Then report back with your numbers.