We can ignore the suits and focus on the suit colours. Since there are only two colours, red and black, we have a binary situation. The probability p=½ that a card in the divided pack would be either a red or a black. So p=q=½ and the binomial expression is:
(p+q)26=1, so, expanding:
p26+26p25q+325p24q2+...+26Crp26-rqr+...+325p2q24+26pq25+q26.
Since p=q=½, 26Crp26-rqr=26Cr/226.
Each term in the expansion represents a probability. p26 or q26 is the probability that all cards in the divided pack will be the same colour. 26C13/226 represents the probability that there are equal numbers of black and red cards. This evaluates to 10400600/67108864=0.155 or 15.5% approximately.
We now need to use Stirling's approximation to see how it compares with the actual probability.
Stirling's formula is:
n!≈√(2πn)(n/e)n and (2n!)≈√(4πn)(2n/e)2n.
2nCn=(2n!)/(n!)2≈2√(πn)(2n/e)2n/(2πn(n/e)2n)=22n/√(πn)).
Therefore 2nCn≈22n/√(πn))=10501063 compared to the actual value 10400600.
And 2nCn/22n≈1/√(πn), and when n=13, this evaluates to about 0.156 or 15.6%.