We have three posibilities
Either
1st round: 6C2 (1/2)^2 (1/2)^4 = 15/64 exactly 2 in the first round,
2nd round:4C0 (1/2)^0 (1/2)^4 = 1/16 exactly 0 in the second round,
3rd round: 4C2 (1/2)^2 (1/2)^2 = 3/8 exactly 2 in the third round.
multiply all 3 rounds we get the first possibility: 15/64X1/16X3/8=45/8192.
OR
1st round: 6C2 (1/2)^2 (1/2)^4 = 15/64 exactly 2 in the first round,
2nd round:4C1 (1/2)^1 (1/2)^3 = 1/4 exactly 1 in the second round,
3rd round: 3C2 (1/2)^2 (1/2)^1 = 3/8 exactly 2 in the third round.
multiply all 3 rounds we get the second possibility: 15/64X1/4X3/8=45/2048.
OR
1st round: 6C2 (1/2)^2 (1/2)^4 = 15/64 exactly 2 in the first round,
2nd round:4C2 (1/2)^2 (1/2)^2 = 3/8 exactly 2 in the second round,
3rd round: 2C2 (1/2)^2 (1/2)^0 = 1/4 exactly 2 in the third round.
multiply all 3 rounds we get the third possibility: 15/64X3/8X1/4=45/2048.
so what you need is the probability that you get exactly 2 in the first round and exactly 0 or 1 or 2 in the second round(no more then this because you need 2 in the third round), and exactly 2 in the third round.The rounds are each (and) functions and we multiply them by each other, and the possibility are an OR function so we add each of the three possibilities
So: 45/8192+45/2048+45/2048=405/8192. tadaaaa