Make sure the auto postback property be set true. This is used to trigger the SelectedIndexChanged event of the dropdownlist
Fill the data for your first dropdown and when the event invoked the code must be like this,
protected void drop_college_SelectedIndexChanged(object sender, EventArgs e)
{
DataSet ds = new DataSet();
SqlDataAdapter dataAdapter = new SqlDataAdapter("Select DepartmentName,DepartmentID FROM DepartmentTable where CollegeName='"+ddlCollegecollege.SelectedItem.Value+"'", connection object);
dataAdapter.Fill(ds);
ddlepartment.DataSource = ds;
ddlDepartment.TextValueField = "DepartmentName";
ddlDepartment.DataValueField = "DepartmentID"
ddlDepartment.DataBind();
ddlDepartment.Items.Insert(0, new ListItem("-- Select Department --", "0"));
}
Hope it helps, Happy coding