A friend says that she will flip a coin 5 times? 
Where did we get 32?
I correctly understood that 10 is a product of 2 and 5 which in turn forms the desired number 32?
 A: *

*$32$ is as stated in the solution of your problem the number of possible outcomes of this game. To count them all you could draw a tree, each layer of the tree representing the possible result from the flip of a coin and all the leaves representing the final 32 possible outcomes (of the type (hhhhh, hhhht, hhhth, hhhtt, etc.)

*$10$ represents the number of way you have, out of those 32 possible outcomes, to get 3 heads (and winning therefore 30$)
Dividing the latter by the former gives you the probability you want to compute.
A: You have 32 possible sequences of outcomes, because you have two values per coin (head vs tail) and 5 independent throws, so $2 \times 2 \times 2 \times 2 \times 2 = 32$ many outcomes in total by the product counting rule. Every coin flip doubles the number of outcomes.
There are $\binom{5}{3}=\frac{120}{6 \times 2}=10$ places where three heads can be thrown vs the two tails. So $10$ of the sequences (HHHTT, HHTHT, HHTTH, HTHHT, etc.) give you $\$30$ payout. So the probability is $\frac{10}{32}=\frac{5}{16}$, a little under a half.
