This question just deals with the charges of the ions.
Long way: Given Y2O3 you know that 2Y + 3O = 0 (neutral compound) Oxygen (as an ion) carries a -2 charge (that is, it will accept 2 e- to get a filled octet). If you plug -2 for oxygen in the equation, the net charge for all 3 oxygens is -6. Seeing that this is a neutral compound you need 2Y to have a +6 charge. So 1 Y ion should carry a +3 charge. If you look on a periodic table, which element has 3 valence electrons it is willing to share to get a filled octet? It can't be Carbon because it has 4 valence electrons, nor can it be Magnesium or Mercury because they only have 2. Ar is a noble gas, which has 8 valence electrons.
Fast way: F- has a -1 charge, and if 3 F- (-3 charge in total) = 1 Y then Y is +3. So which of these can share 3 electrons and get a filled octet in the process? As