We have 3 places to fill, first find all the three digits number containing 7.
Fill 7 at only one place:
7 _ _: we've 2 places to fill, without mixing up more 7s (i.e using these 9 digits 0,1,2,3,4,5,6,8,9), possible numbers are: 9 * 9 = 81
_ 7 _: we've 2 places to fill but here the first digit can't be 0, so 8 digits for first place and 9 digits for the last place excluding 7, possible numbers are: 8 * 9 = 72.
_ _ 7: using the same explanation as above, possible numbers are: 8 * 9 = 72.
Fill 7 at 2 places:
7 7 _: 9 digits possible for the last place excluding 7, so possible numbers = 9
7 _ 7: 9 digits possible for the middle place excluding 7, so possible numbers = 9
_ 7 7: First digit can't be 0, so 8 digits possible excluding 7, so possible numbers = 8
Fill 7 at all 3 places:
7 7 7: possible numbers = 1
Therefore, total numbers containing 7 = 81 + 72 + 72 + 9 + 9 + 8 + 1 = 252
Total numbers between 100 and 999 = 900
Therefore, total numbers not containing 7 = 900 -252 = 648
Therefore, probability that a number selected at random will not contain 7 = 648 / 900 = 18 /25