WASHINGTON — Sandra and William E. Armstrong’s 12-year-old daughter will never learn of the Supreme Court’s deliberations Tuesday, though the results could change her life.

The young Taylorsville, N.C., resident is deaf and blind. She cannot sit, crawl, walk or converse. She’s the victim, her parents say, of negligence by the troubled doctor who delivered her.

She’s at the center of a high-level legal case that pits state governments against the Obama administration and afflicted families. What the high court ultimately decides, after the hourlong oral argument Tuesday, could shape how states stake claims on medical malpractice awards that Medicaid beneficiaries win.

“I think it’s a big accomplishment for our law firm, and for children everywhere,” Sandra Armstrong said of the case in a brief telephone interview. “I’ll be very relieved when it’s over.”

For the Armstrongs, the outcome of the case, known as Delia v. E.M.A., will determine how much of a $2.8 million medical malpractice settlement they can keep. For North Carolina and other cash-strapped states, it could determine how much they can be reimbursed for Medicaid expenses.

E.M.A. are the initials of the Armstrongs’ daughter, Emily. Delia is Albert Delia, the former acting secretary of the North Carolina Department of Health and Human Services.

Underscoring the case’s importance, the 40 million-member AARP is siding with the Armstrongs. So is the U.S. Department of Justice, along with lawyers who file malpractice lawsuits. From the other side, 11 states – including Texas, Idaho and South Carolina – support North Carolina.

The states “have an interest in mitigating the costs associated with administering their Medicaid programs,” Texas Solicitor General Jonathan F. Mitchell noted in the states’ legal brief.

The legal arguments are technical; the facts are horribly poignant.

Emily Armstrong was born Feb. 25, 2000, at what’s now called the Catawba Valley Medical Center, in Hickory, N.C. Sandra Armstrong’s obstetrician, Dr. James A. Barnes Jr., delivered the baby by cesarean section, and it didn’t go well. Emily suffered severe injuries, leading to a diagnosis of cerebral palsy.

The Armstrongs sued Barnes, the medical center and others in 2003. Barnes, it turned out, had a history of drug abuse, including using false prescriptions to obtain pills, according to a subsequent ruling by the North Carolina Court of Appeals. While he denied using illegal drugs at the time of Emily’s prenatal care and delivery, Barnes had voluntarily surrendered his North Carolina medical license in May 2000.

North Carolina state health officials subsequently estimated that they’d spent more than $1.9 million in Medicaid funds providing medical care for the Armstrongs’ daughter.

In a 2006 lawsuit settlement, the Armstrongs received $2.8 million. North Carolina’s former solicitor general, Raleigh-based lawyer Christopher Browning Jr., will argue for the Armstrongs in their effort to fend off the state’s claims on a sizable chunk of the settlement.

“It will help Emily in her needs,” Sandra Armstrong said.

North Carolina officials asserted a lien on $933,333.33, one-third of the total. This claim came under a state law that covers reimbursements from Medicaid beneficiaries who receive money from medical malpractice suits.

The law permits North Carolina to take the lesser of either the total Medicaid spending on the patient or one-third of the court-ordered malpractice payment. In the Armstrongs’ case, because the Medicaid expenditures were so high, the state took the one-third alternative.

After the state staked its claim, the Armstrongs challenged the North Carolina reimbursement law as conflicting with federal law. Here, it gets more complicated.

The federal Medicaid law prohibits state governments from imposing liens on Medicaid patients’ property. A medical malpractice settlement is property. However, a prior Supreme Court ruling specified that the ban on Medicaid liens applies only to the portion of a settlement that doesn’t cover medical care, such as payments for pain and suffering.

The Armstrongs’ settlement is tricky because it doesn’t specify how much is for medical care and how much is for other factors. State officials say they should be the ones who determine how much they can take; by setting the one-third share, for instance.

“The North Carolina Medicaid . . . statutes at issue are part and parcel of the state’s long-standing exercise of its power to oversee tort actions,” North Carolina Solicitor General John F. Maddrey wrote in the state’s legal brief.

States fear, in part, that Medicaid beneficiaries could seek to shield all of their potential medical-malpractice settlements from state claims simply by characterizing all the money as for pain and suffering, rather than medical care. That argument doesn’t satisfy the Armstrongs.

“The threat that states may lose some Medicaid reimbursement funds would provide little comfort to the Medicaid beneficiary who is rendered homeless by a state statute that contravenes the Medicaid Act,” Browning wrote in the Armstrongs’ chief legal brief.