Raising a propertychanged event without adding a private variable

As an alternative to Jon's answer, you can get tools that will do this via IL weaving, such as NotifyPropertyWeaver, also available as a tool through the VS Gallery

For your sample, you should be able to have something like the following, according to their doco on Attributes:

[NotifyProperty(AlsoNotifyFor = new[] { "TotalAnimals" })]
public int NumberOfElephants { get; set; }

public int TotalAnimals { get; set; }

However, based on the example below from their site it might not be required depending on the implementation of TotalAnimals:

Your Code

public class Person : INotifyPropertyChanged {
    public event PropertyChangedEventHandler PropertyChanged;

    public string GivenNames { get; set; }
    public string FamilyName { get; set; }

    public string FullName
    {
        get
        {
            return string.Format("{0} {1}", GivenNames, FamilyName);
        }
    }

}

What gets compiled

public class Person : INotifyPropertyChanged {

    public event PropertyChangedEventHandler PropertyChanged;

    string givenNames;
    public string GivenNames
    {
        get { return givenNames; }
        set
        {
            if (value != givenNames)
            {
                givenNames = value;
                OnPropertyChanged("GivenNames");
                OnPropertyChanged("FullName");
            }
        }
    }

    string familyName;
    public string FamilyName
    {
        get { return familyName; }
        set 
        {
            if (value != familyName)
            {
                familyName = value;
                OnPropertyChanged("FamilyName");
                OnPropertyChanged("FullName");
            }
        }
    }

    public string FullName
    {
        get
        {
            return string.Format("{0} {1}", GivenNames, FamilyName);
        }
    }

    public virtual void OnPropertyChanged(string propertyName)
    {
        var propertyChanged = PropertyChanged;
        if (propertyChanged != null)
        {
            propertyChanged(this, new PropertyChangedEventArgs(propertyName));
        }
    }
}

You don't. You can't.

Automatically implemented propertieS only work when the property is trivial - when no code is needed for the get/set beyond "return the variable's value" or "set the variable's value". You can make it shorter with reformatting, of course... I'd write that as:

private int numberOfElephants;
public int NumberOfElephants {
    get { return numberOfElephants; }

    set {
        _numberOfElephants = value; 
        OnPropertyChanged("totalAnimals"); 
    } 
}

Actually, I'd use "opening brace on a line on its own" for the start of the set and the start of the property, but I've kept your favoured style for those. But having "single expression get/set implementations" on a single line can make classes with lots of properties much cleaner.