Count_Question

A single cell formula devoid of helpers would IMO require use of a Volatile SUMPRODUCT / Array approach and if as you say the dataset is vast that will pretty much lead to your file imploding upon itself in terms of calculation times.

If you don't want to use Concatenation (too many columns) then another (far) simpler route would be to use three calculations adjacent to each row, namely:

=COUNTIF($B2:$F2,24)
=COUNTIF($B2:$F2,48)
=COUNT($B2:$F2) - (or sum of prior 2 calcs)

returning count of 24, 48 and any of 24/48 respectively

From these three cells you can determine the remainder for yourself I think.
(this is a much better route than that advised previously)

If you can't do that then I'm afraid you should really be looking at User Defined Functions (VBA) or simply conduct the calcs on another sheet where you do have space.

Regardless it is important you accept that XL is not a purpose built data warehouse - if you have mass data as implied then you should be using one to store your data and using SQL to analyse.