An experiment consists of tossing a coin ten times and recording the results (heads or tails), for instance, HTHHTHTTTT could represent the result of an experiment. Two results are similar if they differ in no more than two place (or they are identical); thus, HTHHTHTTTT is similar to TTHHTHTHTT, but not to TTTHTHTHTT. Show that in a series of 100 experiments, there will be two with similar results.
I reasoned there are $2^{10}$ possibilities for the sequence of T's and H's. Each sequence will differ by one coin for 10 values. So total similar sequences is $2^{10}-10$. how to put this in terms of $n$ experiments is the tough part I'm running into.also number of ways to choose 1 so similar H or T out of 10 is trivially (10c1) adding that to (10c0) and (10c2) may also be part of the solution but where ? confusion in math is the worst kind
For each result there are ten others that differ in one place. Collect the $100$ sets of eleven, the original result and the ten differing in one place. There are $1024$ possible results and there are $1100$ results among the sets, so there are two results in different sets that match. The achieved results in those two sets are no more than two flips apart.
Added: as an example, let us take series of five flips. Say our first result is $TTTTT$. We group every result within one flip of this, giving $\{TTTTT,HTTTT,THTTT,TTHTT,TTTHT,TTTTH\}$ for six of them. There are $32$ possible results, so if we try six times we will have a match between two of the sets. Maybe another result is $HTTTH$. Its set will be $\{HTTTH,TTTTH,HHTTH,HTHTH,HTTHH,HTTTT\}$ and we note that $HTTTT$ and $TTTTH$ are common to these two sets. Our two results differ in exactly two places, the first and last.