Two column matches typically come in one of two typical configurations; one being an array formula and the other being a standard formula. Both use variations of an INDEX/MATCH function pair.
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The standard formula for Quickbooks!A2 would be,
=IFERROR(INDEX(Donations!A$1:A$999, MIN(INDEX(ROW($1:$999)+((Donations!B$1:B$999<>B2)+(Donations!C$1:C$999<>C2))*1E+99, , ))), "No match")
The array formula version for Quickbooks!A2 would be,
=IFERROR(INDEX(Donations!A$1:A$999, MATCH(1, (Donations!B$1:B$999=B2)*(Donations!C$1:C$999=C2), 0)), "no match")
Array formulas need to be finalized with Ctrl+Shift+Enter? rather than simply Enter?.
Once one of the formulas has been put in Quickbooks!A2 correctly, fill down as necessary to catch the values from other rows. Note that I have changed the third line of your sample data to demonstrate a second DonationID lookup for Check# 106.
With ~50K records, there is the distinct possibility that multiple matches could be made on both columns B and C. To capture the subsequent second, third, etc matches from the Donations worksheet, change the MIN function to a SMALL function and use a progressive COUNTIFS function to adjust the k ordinal.
=IFERROR(INDEX(Donations!A$1:A$50000, SMALL(INDEX(ROW($1:$50000)+((Donations!B$1:B$50000<>B2)+(Donations!C$1:C$50000<>C2))*1E+99, , ), COUNTIFS(B$2:B2, B2, C$2:C2, C2))), "No match")
After setting up some intentional duplicates in the Donations worksheet like this,
DonationID Check# Amount
179 106 $200
210 106 $500
220 106 $600
979 106 $200
910 106 $500
920 106 $600
You should receive results like the following.
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The IFERROR function has been used to catch non-matches and show no match rather than the #N/A
error.
I have modified the values in your single worksheet sample data to show a variety of matches as well as one case where the chknum and amount were duplicated by two different donationID entries.
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The matching records are color-coded for quick reference. I've made this sample workbook publically available at my Docs.com-Preview site.
?VLOOKUP_w_Multiple_Criteria_and_Duplicates.xlsx