I assume the usual (though unstated) provisions that each letter stands for a different integer (0..9), and that none of the four "words" starts with zero.

There are two numbers for which a single solution:

for Y=2 -- 65 + 735 + 9532 = 10332

for Y=3 -- 75 + 625 + 9523 = 10223

For two other values of Y there are two solutions each:

for Y=4 -- 75 + 625 + 9524 = 10224

for Y=4 -- 65 + 735 + 9534 = 10334

for Y=8 -- 75 + 625 + 9528 = 10228

for y=8 -- 65 + 735 + 9538 = 10338.

+ 1