How many 6 card hands have at LEAST two hearts and at LEAST two spades? Using a standard deck. Thanks in advance!?

You need to break it up into cases with exactly how many hearts, spades and other cards there are.

If there are h hearts, s spades, and o others, with h and s always >= 2 and h+s+o always = 6, then you have:

13Ch X 13Cs + 26Co.

If you try to calculate it another way, like assuming you have 2 hearts, 2 spades, and 2 others from the other 48 cards, you overcount them.

There really aren't that many cases to work out. Only 6. Here are the results:

h s o 13Ch 13Cs 26Co Product

2 2 2 78 78 325 1,977,300

2 3 1 78 286 26 580,008

2 4 0 78 715 1 55,770

3 2 1 286 78 26 580,008

3 3 0 286 286 1 81,796

4 2 0 715 78 1 55,770

Total 3,330,652

Sorry Y!A doesn't align columns, but it should be clear. The answer is 3,330,652

I am afraid you will have to do a bit of adding to count

[ {4H,2S}/{2H/4S} or {3H,3S} ] + 0 others

[ {3H,2S}/{2H,3S} ] + 1 others

[ {2H,2S} ] + 2 others

[2*C(13,4)*C(13,2) + (C(13,3))² + 2*C(13,3)*C(13,2)*C(26,1) + (C(13,2))² *C(26,2)] = 3,330,652 <------