Well, what model do you pass to your partial view?
Reading your question it sounds like you need a model which tells you what is selected then you can,
in the partial view, set the initial states of the check boxes.
In turn, depending on the rest of your view/page/site put in jQuery so that on the client side if they change the "transport" the appropriate passenger changes.
---------Below is based on code provided-------------
This is sample code it may need tweaking but should give you sufficient to get started. It is also assumed that you are surrounding all of your view in a "Form" which submits to an action expecting your model. It also assumes you are using bootstrap classes (as it appears you might be so the styling might be slightly out).
@Html.Label("Vehicle", new { @class = "control-label" })
</div>
<div class="col-md-3">
<label class="radio-inline">
<input type="radio" name="Car" id="inlineCar" value="Car" @(Model.Car.Equals("Car") ? "Checked" : String.Empty)> Car
</label>
<label class="radio-inline">
<input type="radio" name="Car" id="inlineBus" value="Bus" @(Model.Car.Equals("Bus") ? "Checked" : String.Empty)> Bus
</label>
<label class="radio-inline">
<input type="radio" name="Car" id="inlineTrain" value="Train" @(Model.Car.Equals("Train") ? "Checked" : String.Empty)> Train
</label>
@Html.ValidationMessageFor(model => model.Car)
</div>
Do the same for your number of seats, but also have a check to see if you need to disable any of the radio buttons.
Then you need to write some jquery code that works when the vehicle is changed to update the selected number of seats.
Hope this gives you enough to accomplish your objective.