Combinatorics

How many ways we can have AA in our hand? 
AcAs, AcAd, AcAh, AsAd, AsAh, AdAh – total 6 possible combinations.

How many ways we can have AK? – 16 (you can count yourself).

Now more interesting question: How many ways our opponent can have AA when we have Ad in our hand? 
AcAs, AcAh, AsAh – total of 3 possible combinations.
So when we were dealt an Ace chances that our opponent was dealt two Aces are less by 50% Our Ace blocks half of his combinations. We can say we have a ‘blocker’ to AA.

How to use this knowledge on practice?
Say our tight opponent open raises preflop from early position. We think he can have AA, KK, QQ, JJ, AK or AQ. When we look at our hand we see that we were dealt AK. Now we can deduce distribution of hands in our opponent’s range. He has 3 combos of AA, 3 of KK, 6 of QQ, 6 of JJ, 9 of AK and 12 of AQ for a total of 39 combinations. 
From here we know that he has: 
AA 3 out of 39 - 8%,
KK 3 out of 39 - 8%,
QQ, JJ, AK 21 out of 39 - 54%,
AQ 12 out of 39 – 30% of time.

Now let’s calculate our equity versus his range:
AK has 8% vs AA,
30% vs KK,
50% vs QQ, JJ and AK,
70% vs AQ.

For a total of 8 * 0.08 + 30 * 0.08 + 50 * 0.54 + 70 * 0.30 = 51% equity. 

So we should expect to win about half of the times we go all-in preflop versus this opponent.