CLOSE PROX BETWEEN ACR AND COMMUTER ACFT IN TCA. OPERATIONAL ERROR.

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WE; ACR X; DEPARTED TO S ON THE DCA 185 DEG R CLBING TO 17000'. MY F/O ALERTLY ADVISED ME THAT WE HAD TFC AT 12 O'CLOCK. IT WAS HEAD-ON AND ON A COLLISION COURSE. AN IMMEDIATE EVASIVE TURN TO THE LEFT WAS MADE. THE ONCOMING TFC DID NOT APPEAR TO REACT TO US AND PASSED SLIGHTLY BENEATH AND WITHIN 500' ESTIMATED TO OUR RIGHT. ELAPSED TIME FROM SIGHTING TO PASSAGE MAY HAVE BEEN LESS THAN 15 SECS. WHEN ASKED WHO JUST PASSED US; DEP CTL TENTATIVELY IDENTED THE ACFT AS COMMUTER Y. WE WERE ASKED TO CYCLE OUR TRANSPONDER AND THEN TO TURN TO HDG 030 DEGS FOR IDENT. HE STATED THAT ACFT Y WAS ON APCH CTL AND HAD HAD US IN SIGHT. WE DOUBLE-CHKED OUR TRANSPONDER SWITCHES AND CONFIRMED THAT THEY WERE PROPERLY POSITIONED. THE CAPT WAS AT THE CTLS OF OUR ACFT AT THE TIME OF THE INCIDENT. THE EVASIVE TURN WAS SMOOTH AND THERE WERE NO INJURIES TO CREW OR PAX. NO SUBSEQUENT CTLR INDICATED ANY DIFFICULTY RECEIVING OUR TRANSPONDER. USING THE SAME AIRSPACE CORRIDOR FOR DEPS AND ARRS ON SEPARATE FREQS CONTRIBUTED TO THIS NEAR ACCIDENT. BOTH ACFT HAD LNDG LIGHTS ON WHICH AIDED IN OUR ABILITY TO SEE AND AVOID. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING: SINCE INCIDENT RPTR HAS RECEIVED PHONE CALL FROM DCA FAC MGR IN WHICH HE ADMITTED ATC ERROR. SAYS THE DEP CTLR GOT BUSY TRYING TO ASSIST ANOTHER RADAR CTLR AT ADJACENT POS AND FAILED TO TURN ACR X AWAY FROM COMMUTER Y ACFT. RPTR SURE THAT THEY HAD RECEIVED CLRNC FOR CLB TO 17000'; AND WERE CLBING RAPIDLY DUE TO LOW GROSS WT. THINKS ACFT Y DID NOT USE GOOD JUDGEMENT IN FLYING SO CLOSE TO ACR X IF THEY HAD TFC IN SIGHT AND WERE TO PROVIDE VISUAL SEP. DOES NOT KNOW IF ACFT Y WAS IFR OR VFR FLT. GLAD THAT ACFT HAD LNDG LIGHTS ON.