Based on your information - you was a beneficiary of the trust that was set by your mother.
It is important to know if the trust was set as revocable or irrevocable.
It is also important to know if there was any tax deferred accounts - such as 401k, IRA, etc that your mother put into the trust.
Please specify how the IRS determined that your capital gain was $15,000?
Did you received any reporting forms that indicated taxable income - may be some came in your mother's name?
As the trust was established by your great grandmother - it became irrevocable after she died - so you do not own the trust and the trust is not considered your inheritance - but you became a trustee and a beneficiary since your mother died.
As the trust owned some shared of mutual funds - and they were liquidated - the trust might recognize the capital gain. To calculate the capital gain - you need to know the basis of each share and the sale price - the difference would be a capital gain.
In additional the trust might received some dividends and capital gains that are also taxable income.
Depending how the trust was set - either the trust pay income tax on the income - or income gets distributed to beneficiaries. The distributed income is reported on the schedule K-1 - that form you should provide to your tax preparer along with other income reporting documents.
I believe that schedule K-1 reports that $15,000 income which the trust distributed to you and that should be added to your taxable income.
As the schedule K-1 is also sent to the IRS - the IRS found the discrepancy with your personal tax return and sent a notice.
Please verify if my assumptions are correct - and if so - you do not have a choice but to pay additional taxes.
Based on information you provided assets were not owned by your mother but were owned by the irrevocable trust that is a separate entity.
So you may not treat these assets as inherited - but you received them as a beneficiary of the trust.
If assets were owned by your mother and inherited by you - that will be a different situation from tax prospective.
As assets in the trust were liquidated - that what triggered capital gains - and my assumption is that the gain was passed through to you as a beneficiary. I suggest that you verify taxable amounts reported on the schedule K-1.
It might be too late - but if assets were transfered to you without being sold - there would not be any taxable gain.
You may not own the trust - you likely became a trustee and a beneficiary of the trust - but the distribution from the trust may be partially taxable.
It is not clear if your inherited cash or mutual fund shares - inheritance is not taxable - so whatever you inherited from your mother is not included into your income tax return.
Any income received in that account before your mother died - interest, dividends, etc - should be reported on her final return.
However if you inherited mutual fund account - and sell shares - you may recognize capital gain - if any case - you meed to report the sale transaction on the schedule D.
Your gain = (selling price) - (basis), where the basis for inherited property is a fair market value on the day your mother died.
You need to report the sale transaction and determine the basis - if you do not report - the IRS would assume the gross sale amount is your income.
If you sold shortly after your mother's death - it is likely - no or very small gain.
If you inherited tax deferred mutual fund account - like IRA, 401k, etc - it is taxable when you take distribution. You may spread distribution over several years to reduce tax pressure.