One problem - two approaches, two different answers

A test is based upon the average of 5 results. Assuming only random errors in measurement, what is the probability the data is trended (sequential up or down). So 5 results can be distributed 120 ways - and there are only two trends – “12345” and “54321”. So we have a 2/120, p=0.0170, chance of seeing a “Trend” when there is no trend.
Using a different approach, the next measured result can be higher or lower (p=0.5) than present result. Do this four times to fill the 5 result test, p=(0.5)^4 or p=0.0625 overall. Since there are 2 cases: 5 ascending and 5 descending, the probabilities sum to p=0.125.
So how can one reconcile the two, 1/60 does not equal 1/8 ?

In 30 tests I see two trended data sets.... is my test unbiased?