I've been re-reading my probability books for the math we're doing with ofc. I thought it would be good to look at some hold'em stuff too.

How does the math show that there are 169 TYPES of starting hands in texas hold'em?

SHORT ANSWER

There are 13 types of pairs.

Ignoring suits, there are C(13,2) types on non-pairs. That comes to 13!/11!2! or 13*12/2 or 78. Factoring in suits, we have 78 suited non-pairs and 78 non-suited non-pairs.

13 types of pairs
78 types of suited non-pairs
78 types of non-suited non-pairs
-----------------------------------
169 TYPES of starting hands

LONG ANSWER

There are C(52,2) starting hands. That comes to 52!/49!2! or 52*51/2 or 1,326.

Of course this counts A A and A A as two different hands so we need to do some consolidating to get to 169 TYPES of starting hands. We know that each of the 13 pairs can be made C(4,2) or 4!/2!/2! or 4*3/2 or 6 ways so there are 13*6 or 78 pairs.

Here are the 6 pairs of aces:
A A
A A
A A
A A
A A
A A

We can treat all 6 of these pairs of aces the same. Consolidating, we have 78/6 or 13 TYPES of pairs.

That leaves 1,326 - 78 or 1,248 non-pairs.

Let's look at AK.

There are 4 ways to have AK suited:
A K
A K
A K
A K

There are 12 ways to have AK offsuit:
A K
A K
A K
A K
A K
A K

K A
K A
K A
K A
K A
K A

This means we have 3 times as many offsuit hands as suited hands for the 1,248 non-pairs. Let x equal suited non-paris. x + 3x = 1,248 which means 4x = 1,248 which means x = 1,248/4 which means we have 312 suited non-pairs. That leaves 936 offsuit non-pairs.

Consolidating the 312 suited hands, we know we can treat each of the 4 suits the same so we divide by 4. This means we have 312/4 or 78 TYPES of suited hands.

Consolidating the 936 offsuit non-pairs, we know we can treat all 12 of the above AK offsuit hands the same. This is the same for all 936 hands so we have 936/12 or 78 TYPES of offsuit non-pairs.

We're left with the same totals as the short answer:
13 types of pairs
78 types of suited non-pairs
78 types of non-suited non-pairs
-----------------------------------
169 TYPES of starting hands

As you can see, it is more likely to get some of the 169 TYPES of hands than others. AK offsuit is more likely than AA which is more likely than AK suited.

*Note that we used combinations and not permutations because we're not concerned about the order in which the two hole cards were dealt. In other words, we treat A 3 the same whether we were dealt the ace first or the trey first.