Well, the key to finding T is to count how many powers of 10 (ie,2*5) are in the factorial. Since there are at lot more 2's than 5's, all we need to do is count the powers of 5.

The given factorial is equivalent to: 100*99^2*98^3*97^4 ...2^99

For purposes of counting 5's, all we need are the factors that include a 5: 100*95^6*90^11*...5^96.

factor x contributes (101-x) 5's, unless it is a multiple of 25, in which case it contributes 2*(101-x) 5's. My tally is

100 2 <-- multiple of 25

95 6

90 11

85 16

80 21

75 52 <-- multiple of 25

70 31

65 36

60 41

55 46

50 102 <-- multiple of 25

45 56

40 61

35 66

30 71

25 152 <-- multiple of 25

20 81

15 86

10 91

5 96

----

T = 1124, so the requested answer is 124