Probability Question

If three cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the three cards if at least two cards are a jack, queen or a king?

Total cards in a deck = 52(13 of each of the 4 types)
Total kings =4
Total queens = 4
Total jacks =4
4+4+4 = 12

Now there are 2 possibilities

1) 2 cards are either a king,queen or a king
2) All 3 cards drawn are either a king,queen or jack

1) 12C2*40C1 = 6*11*40 = 2640

2) 12C3 = 2*10*11 = 220

Hence total = 2640+220 = 2860

I got the same answer but the answer given is 2925. Just wanted to cross check.

Cybermusings wrote: If three cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the three cards if at least two cards are a jack, queen or a king?

Now there are 2 possibilities

1) 2 cards are either a king,queen or a king
2) All 3 cards drawn are either a king,queen or jack

1) 12C2*40C1 = 6*11*40 = 2640

2) 12C3 = 2*10*11 = 220

Hence total = 2640+220 = 2860

I have general question here. Given that we are asked to find the
total number of ways in which the cards can be chosen, is it correct
to use 12C2 -- shouldn't it be 12P2, since say KQ2 and QK2 are
different ways of "drawing" the 3 cards.

More specifically, going by how the draw takes place --

Prob of choosing 2 paint-cards = 12 * 11 * 40 (which is the same as 12P2 * 40)

Can someone please confirm.