The Indiana State Archives (e-mail firstname.lastname@example.org
) has the records of the Indiana Boys School in Plainfield. They do have a database for boys admitted to the school from 1867 to 1937 (http://www.indianadigitalarchives.org/TitleInfo.aspx?TID=128
) but it doesn't show any BINGAMAN/BINGMAN entries.
I do see his October 23, 1918 death is listed in the WPA's death index for Hendricks County. You can request a copy of his death record from the Indiana State Department of Health (http://www.in.gov/isdh/20444.htm
), as that will have more information than the death card you can get from the Hendricks County Health Department.
You may also want to look for mention of his death in a Cass County or Hendricks County newspaper, to see if it mentions anything about how he died. The Indiana State Library has microfilm of newspapers from all over the state - your local library can request them via inter-library loan (see http://www.in.gov/library/newspapers.htm