p=0.20 is the probability that Americans think that N is the greatest constituent of Earth's atmosphere, so 0.80 Americans think not. The distribution is binomial:
(0.20+0.80)^6=1(certainty)=0.2^6+6*0.2^5*0.8+15*0.2^4*0.8^2+20*0.2^3*0.8^3+15*0.2^2*0.8^4+6*0.2*0.8^5+0.8^6,
where each term gives the combined probabilities for exactly 6, 5, 4, ... of p combined with 0, 1, 2, ... of (1-p).
So the probability of exactly 2/6 identifying N is the 5th term 15*0.2^2*0.8^4=0.24576.