Tom, Dick and Harry each chose a different 5-digit

triangular number whose digits fell into the pattern THREE, where each different letter represents a different base-10 digit.

Dick's triangular number had no digit in common with either Tom's or Harry's.

What was Dick's number? What two triangular numbers did the other two choose?

There are nine triangular numbers which meet this pattern (assuming no leading zero). Dick's number is between the other two, but the sum of the five digits in his is greater than the sum of the ten digits in the other two. Cf NS1510 (I peaked)