a) How many three-digit multiples of 5 have three different digits?

a):  

105 , 120 , 125 , 130 , 135 , 140 , 145 , 150 , 160 , 165 , 170 , 175 , 180 , 185 , 190 , 195 , 205 , 210 , 215 , 230 , 235 , 240 , 245 , 250 , 260 , 265 , 270 , 275 , 280 , 285 , 290 , 295 , 305 , 310 , 315 , 320 , 325 , 340 , 345 , 350 , 360 , 365 , 370 , 375 , 380 , 385 , 390 , 395 , 405 , 410 , 415 , 420 , 425 , 430 , 435 , 450 , 460 , 465 , 470 , 475 , 480 , 485 , 490 , 495 , 510 , 520 , 530 , 540 , 560 , 570 , 580 , 590 , 605 , 610 , 615 , 620 , 625 , 630 , 635 , 640 , 645 , 650 , 670 , 675 , 680 , 685 , 690 , 695 , 705 , 710 , 715 , 720 , 725 , 730 , 735 , 740 , 745 , 750 , 760 , 765 , 780 , 785 , 790 , 795 , 805 , 810 , 815 , 820 , 825 , 830 , 835 , 840 , 845 , 850 , 860 , 865 , 870 , 875 , 890 , 895 , 905 , 910 , 915 , 920 , 925 , 930 , 935 , 940 , 945 , 950 , 960 , 965 , 970 , 975 , 980 , 985 , Total =  136 such 3-digit numbers.

b):

130 , 135 , 150 , 170 , 175 , 190 , 195 , 210 , 215 , 230 , 235 , 250 , 270 , 275 , 290 , 295 , 310 , 315 , 350 , 370 , 375 , 390 , 395 , 410 , 415 , 430 , 435 , 450 , 470 , 475 , 490 , 495 , 510 , 530 , 570 , 590 , 610 , 615 , 630 , 635 , 650 , 670 , 675 , 690 , 695 , 710 , 715 , 730 , 735 , 750 , 790 , 795 , 810 , 815 , 830 , 835 , 850 , 870 , 875 , 890 , 895 , 910 , 915 , 930 , 935 , 950 , 970 , 975 , Total =  68 such 3-digit numbers

a) If the ones digit is a 5, there are 8 choices for the hundreds digit (can't be 0 or 5) and 8 choices for the tens digit. So there are \(8 \times 8 = 64\).

    If the ones digit is a 0, there are 9 choices for the hundreds digit and 8 for the tens digit which makes for \(9 \times 8 = 72\) choices

So, there are \(64 + 72 = \color{brown}\boxed{136}\) numbers 

b) Half of the numbers have an odd tens digit, so there are \(136 \div 2 = \color{brown}\boxed{68}\)