Sumproduct or Countif on a 2D matrix - excel

Question

I'm working on data from a population of people with allergies. Each person has a unique ExceptionID, and each allergen has a unique AllergenID (451 in total).
I have a data table with 2 columns (ExceptionID and AllergenID), where each person's allergies are listed row by row. This means that the ExceptionID column has repeated values for people with multiple allergies, and the AllergenID column has repeated values for the different people who have that allergy.
I am trying to count how many times each pair of allergies is present in this population (e.g. Allergen#107 & Allergen#108, Allergen#107 & Allergen#109,etc). To keep it simple I've created a matrix of 451 rows X 451 columns, representing every pair (twice actually because A/B and B/A are equivalent).
I somehow need to use the row name (allergenID) to lookup the ExceptionID in my data table, and count the cases where that matches the ExceptionIDs from the column name (also AllergenID). I have no problem using Vlookup or Index/Match, but I'm struggling with the correct combination of a lookup and Sumproduct or Countif formula.
Any help is greatly appreciated!
Mike
PS I'm using Excel 2016 if that changes anything.
-=UPDATE=-
So the methods suggested by Dirk and MacroMarc both worked, though I couldn't apply the latter to my full data set (17,000+ rows) because it was taking a long time.
I've since decided to turn this into a VBA macro because we now want to see the counts of triplets instead of pairs.

Answer 1

With the 2 columns you start with, it is as good as impossible... You would need to check every ExceptionID to have 2 different specific AllergenID. Better use a helper-table with ExceptionID as rows and AllergenID as columns (or the opposite... whatever you like). The helper table needs a formula like:
=COUNTIFS($A:$A,$D2,$B:$B,E$1)
Which then can be auto-filled. (The ranges are from my example, you need to change them to your needs).
With this helper-matrix you can easily go for your bigger matrix like this:
=COUNTIFS(E:E,1,INDEX($E:$G,,MATCH($I2,$E$1:$G$1,0)),1)
Again, you can auto-fill with this formula, but you need to change it, so it fits your needs.
Because the columns have the same ID2 (would be your AllergenID), there is no need to lookup them because E:E changes automatically with the auto-fill.
Most important part of the formulas are the $ which should not be messed up, or you can not auto-fill it.
Picture of my self-made example (formulas are from the upper left cell in each table):
If you still have any questions, just ask :)

Answer 2

It can be done straight from your original set-up with array formulas:
Please note that array formulas MUST be entered with Ctrl-Shift-Enter, before copying across and down:
In the example pic, I have NAMED the data ranges $A$2:$A$21 as 'People' and $B$2:$B$21 as 'Allergens' to make it a nicer set-up. You can see in the formula bar how that looks as a formula. However you could use the standard references like this in your first matrix cell:
EDIT: silly me, N function is not needed to turn the booleans into 1's and 0's, since multiplying booleans will do the trick. Below formula works...
SUM(IF(MATCH($A$2:$A$21,$A$2:$A$21,0)=ROW($A$2:$A$21)-1, NOT(ISERROR(MATCH($A$2:$A$21&$E2,$A$2:$A$21&$B$2:$B$21,0)))*NOT(ISERROR(MATCH($A$2:$A$21&F$1, $A$2:$A$21&$B$2:$B$21,0))), 0))
Then copy from F2 across and down. It can be perhaps improved in technique with sumproduct or whatever, but it's just a rough example of the technique....