Note that the expansion of (3√x-2 (opened three times) √ x)11+0 is 12.
And its term is c (12, r) (3 √ x) (12-r) (-2x (1/3)) r.
=c( 12,r)(3)^( 12-r)(-2)^rx^( 12-r)/2x^(r/3)
=c( 12,r)(3)^( 12-r)(-2)^rx^[( 12-r)/2+(r/3)]
=c( 12,r)(3)^( 12-r)(-2)^rx^[( 12-r)/2+(r/3)]
=c( 12,r)(3)^( 12-r)(-2)^rx^[(36-r)/3)]
When r = 0, 3, 6, 9, 12, these terms are rational numbers.
So there are five rational terms * * *,
Therefore, the probability that the selected item is rational is a =5/ 12.