The probability that in a random arrangement of letters of the word "COLLEGE" so that neither 2E's nor 2L's come together is

A is the event of 2 E’s come together

$P\left(A\right)=\frac{\frac{6!}{2!}}{\frac{7!}{2!2!}}=\frac{2}{7}$

B is the event of 2 L’s come together

$P\left(B\right)=\frac{\frac{6!}{2!}}{\frac{7!}{2!2!}}=\frac{2}{7}$

$P\left(A\cap B\right)=\frac{5!}{\frac{7!}{2!2!}}=\frac{2}{21}$

$P\left(\overline{A\cup B}\right)=1-P\left(A\cup B\right)$

$=1-\left[P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\right]$

$=\frac{11}{21}$