How many three-digit numbers are there in which all three digits are different and the number itself is odd?

Deal with the tightest place first. The units digit carries the odd condition, so it is 1, 3, 5, 7 or 9 — 5 choices. The hundreds digit may not be 0 and may not repeat the units digit, which leaves 8 choices. The tens digit may be anything except the two digits already used, so 8 choices. Total = $5\times 8\times 8=320$. Trap $360=5\times 9\times 8$ forgets that a three-digit number cannot begin with 0. Trap $405=5\times 9\times 9$ lets the hundreds and tens digits repeat each other. Trap 450 is half of all 900 three-digit numbers and uses the odd condition alone, ignoring 'all different'.