All arrangements of the seven letters of the word BALLOON (using all letters) are considered.

The total number of arrangements is $\dfrac{7!}{2!\,2!}=1260$. (A) Gluing the two O's into one block leaves 6 objects with the two L's identical: $\dfrac{6!}{2!}=360$; true. (B) Arrangements with the L's together also number $360$, so those with L's apart are $1260-360=900$; true. (C) By inclusion-exclusion, subtract the arrangements with the O's together ($360$) and with the L's together ($360$), then add back those with both pairs glued, $5!=120$: $1260-720+120=660\ne600$; false. (D) Among the relative orders of A, O, O, the letter A is first in exactly one of three equally likely positions, so the count is $1260/3=420$; true. A brute-force enumeration of the distinct permutations confirms all four counts.