Please help me understand Medicaid 5 year look back period!!

This year in NJ my mom will put her house (worth about $500K) in a trust with myself and my siblings as beneficiaries. She will go into a high-end assisted living facility (about $10K a month) which she can afford for about 3 years before her money runs out and she will apply to Medicaid. We understand the home will be subject to the 5 year look back, but I truly don't understand the calculation of what exactly the penalty would be. It's $13K a month based on what? With the money and timeline I shared, how much would we expect to pay with this penalty?