A coin is flipped 10 times where each flip comes up either heads or tails....

Step 1
Definitions
Product rule if one event can occur in m ways AND a second event can occur in n ways, then the number of ways that the two events can occur in sequence is then m*n.
Definition permutation (order is important):
${\displaystyle P(n,r)=\frac{n!}{(n-r)!}}$
Definition combination (order is not important):
$$C(n,r)=(\begin{array}{c}n\\ r\end{array})=\frac{n!}{r!(n-r)!}$$
with n!=n*(n-1)*...*2*1.
Step 2
The order of the heads/tails is not important (since we are interested in the number of heads, not the order of the heads), thus we need to use the definition of combination.
An equal number of heads and tails in 10 flips, means that there are five heads and five tails.
n=10
r=5
Evaluate the definition of a combination:
${\displaystyle C(10,5)=\frac{10!}{5!(10-5)!}=\frac{10!}{5!5!}=252}$