AMS employs 8 professors on their staff. Their respective probability

Sure! Its not a GMAT test question.....But "or any other time constrained test" is falsely assumed... It is a question from CAT , which is an mba entrance test in india... Anyways.. I posted on this forum for the sake of learning and also because this forum has fastest replies of all... Kudos for that....

So, My question is that is there any faster way of solving such a question with multiple different probabilities.... can we take average of probabilities in any case whatsoever?

here 8 professors are there.
Asking atleast 6 of them continue ,
it has 3 cases.
1 all 8 professors continue.
2 7 professors continue and 1 professors discontinue.
3 6 professors continue and 2 professors discontinue.

1st case all 8 continue is = 2/10*3/10*4/10*5/10*6/10*7/10*/8/10*9/10=9factorial/10power8
=362880/100000000
=>0.00363.

2nd case any 7 professors continue, and 1out of 8 discontinue ,8C1 means 8 ways.
= 2/10*3/10.......8/10*1/10, (9/10 prodbability professor discontinue then 1/10)
in this way if we calculate for 8 possibilities then value is =>0.03733.


3rd case any 6 professors continue and 2out of 8 discontinue , 8C2 means 28 ways.

= 2/10*3/10.....2/10*1/10( 2 professors discontinue.

if we calculate for 28 possibilites P value is=>0.1436

=0.00363+0.03733+0.1436=0.18456=
0.19
Option A sufficient.