incorporating share buyback in entry multiple

Hi guys. Would like to seek your wisdom on hypothetical valuation case.

Assume that sponsor buys 40% stake of the target at 250. The target has net cash of 200 and recorded LTM EBITDA of 50. The purchase price is fixed as this comes after negotiation with the seller. Right after the acquisition, the target buybacks entire shares remaining in the market (60%) as treasury shares and takes the company private. This cost the target 200. (This might not be the case in U.S. as you might be obligated to buy other shareholders stake at the same price. However, disregard this regulation) In this case what would be the entry EV/EBITDA multiple for the sponsor?

1) 100% implied equity value: 250 (for the 40%) + 200 (for the 60%) = 450 Net Cash: 200-200=0 EV = 450 +0 =450 EV/EBITDA = 9x

2) 100% implied equity value: 250 (considering the effective ownership which is 100%, as the target will cancel out the treasury shares) Net Cash: 200-200=0 EV = 250+0 =250 EV/EBITDA = 5x

Which one is right? 1) or 2)? If 2) is confusing, let's approach in this way. The sponsor requires the target to buyback 60% stake as treasury shares and the target cancels treasury shares prior to the transaction with the sponsor. As a result, sponsor will be able to purchase 100% equity at 250, and the company will have 0 net cash.

Thanks,