Letters testamentary are issued by the probate court to a person appointed to be personal representative of an estate under a will. Letters of administration are issued when there is no will (when the person died intestate). There is a statutory scheme that gives priority as to who to grant letters of administration to (e.g., spouse, children, siblings, etc.).
Unfortunately, it is sometimes the circumstance where an individual will pass away testate and the individual(s) appointed under the will are also deceased. In such circumstances, letters testamentary cannot be issued because an appointee is not alive and typical letters of administration cannot be issued because the individual died with a will.
Enter the letters of administration with will annexed (sometimes called by its Latin name cum testamento annexo). When an individual passes away testate and no appointed personal representative is alive or willing to serve, then letters of administration with will annexed will be granted to a duly qualified individual under Missouri law.
There are also the rare circumstances where letters of administration with good un-administered (de bonis non) may be issued. This is appropriate when a person is appointed personal representative and either dies, resigns, or is removed; the successor — whoever that may be — will have to finish administering the estate.