Conditiona Probability?

In a bag, there are 5 white balls, 7 red balls and 8 black balls. Three balls are chosen randomly in succession from the bag, one after one. If the first and the third chosen balls are found to be in different colours, find the probability that the second chosen ball is red.
Sol
(1) the first is white and the third is red
P1=(5/20)*(7/19)*(6/18)=210/6840
(2) the first is white and the third is black
P2=(5/20)*(8/19)*(7/18)=280/6840
(3) the first is red and the third is white
P3=(7/20)*(5/19)*(6/18)=210/6840
(4) the first is red and the third is black
P4=(7/20)*(8/19)*(6/18)=336/6840
(5) the first is black and the third is white
P5=(8/20)*(5/19)*(7/18)=280/6840
(6) the first is black and the third is red
P6=(8/20)*(7/19)*(6/18)=336/6840
P=P1+P2+P3+P4+P5+P6
=(210+280+210+336+280+336)/6840
=413/1710