LTT ACFT LANDED ON RECIPROCAL RWY AFTER BEING CLRED TO LAND. FLC FLEW THE APCH FOR THE OTHER RWY AND LANDED.

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ENRTE TO SWF FROM THE W; WE WERE TOLD BY NY APCH TO EXPECT VOR RWY 27 APCH. WE REQUESTED ILS RWY 9 APCH AND WERE TOLD TO EXPECT ILS RWY 9; THEN RECEIVED OUR FIRST VECTOR TOWARD INTERCEPTION. AS WE NEARED THE FINAL APCH FIX; WE WERE GIVEN A NEW HDG 'FOR THE VOR RWY 27 APCH.' WE BRIEFED THE APCH AND CONTINUED ON; FLYING THE VOR RWY 27. AT SOME POINT DURING THE COMS (SAME CTLR); WE WERE ASKED IF WE HAD REQUESTED THE NDB FOR RWY 9; AND THAT HE WAS SORRY WE WEREN'T GIVEN THAT; THAT HE HADN'T GOTTEN THE WORD (PERHAPS A CHANGE IN PERSONNEL ON THE SAME FREQ). I REPLIED; 'NO; I HAD REQUESTED TO ILS.' HE APOLOGIZED AND TURNED US OVER TO SWF TWR. TWR CLRED US TO LAND RWY 9; WE LANDED WITHOUT INCIDENT ON RWY 27 AND IT WAS ONLY AFTER TURNING OFF THE RWY THAT TWR QUERIED US ASKING US WHICH APCH WE WERE GIVEN. HE SAID HE HAD BEEN TOLD WE WERE ON THE ILS RWY 9. I REPLIED WE HAD REQUESTED ILS RWY 9; BUT WE HAD BEEN GIVEN VOR RWY 27. IT WAS THEN HE SAID THAT HE HAD CLRED US TO LAND ON RWY 9 (ONLY THEN DID WE REALIZE WE WERE CLRED TO LAND/LANDED ON THE WRONG RWY). HE SAID IT WAS NO PROB. CAUSE OF PROB: 1) INCORRECT COM BTWN APCH CTLR AND TWR. 2) CREW DID NOT REALIZE TWR CLRED US TO LAND ON THE WRONG RWY AND DID NOT READ ENTIRE CLRNC BACK TO INCLUDE RWY NUMBER. SOLUTION: 1) BETTER COM BTWN SHIFT CHANGE CTLRS. 2) BETTER COM BTWN APCH AND TWR CTLRS. 3) HEAR AND READ BACK FULL CLRNC INCLUDING RWY NUMBER (WE HEARD RWY NUMBER WE WANTED/EXPECTED TO HEAR).