Concept:We count 4-digit odd numbers (1000–9999) with no repeated digits by considering the unit digit (must be odd) and the thousand's digit (cannot be zero).
Explanation:Step 1: The unit digit must be odd: 1, 3, 5, 7, 9 — 5 choices.
Step 2: The thousand's digit cannot be zero and must differ from the unit digit.
We split into cases to avoid double‑counting when zero appears in other positions.
Case A: No digit is zero (thousand's digit is from 1–9 except the chosen unit digit, and the other two digits are chosen from remaining non‑zero digits).
Choose unit digit: 5 ways.
Choose thousand's digit: from 1–9 excluding unit digit ⇒ 8 possibilities.
Choose hundred's digit: from the remaining 8 digits (including zero? here zero is excluded because we are in case "no zero") ⇒ actually after picking unit and thousand, 8 non‑zero digits remain? Wait, we have digits 1–9, we used one for unit, one for thousand → 7 non‑zero left, plus zero? But case says no zero, so only non‑zero. So 7 choices for hundred's? The original solution says 7 choices for hundred's in case 1? Let's recalc: total digits 0–9 (10 digits). Unit odd (5 choices). Thousand cannot be zero and not same as unit: from 1–9 excluding unit → 8 choices. Now we have used 2 digits, remaining 8 digits include zero and 7 non‑zero. In case "all digits non‑zero", we restrict to non‑zero only: so for hundred's we choose from the 7 non‑zero digits not used. Then for ten's, choose from remaining 6 non‑zero digits. So count = 5 × 8 × 7 × 6 = 1680. That matches the solution.
Case B: Hundred's digit is zero. (The number has a zero in the hundreds place; thousand's and unit are non‑zero, ten's is any digit except used ones).
Unit digit: 5 odd choices.
Thousand's digit: cannot be zero and not equal to unit ⇒ 8 choices (from 1–9 excluding unit).
Hundred's digit: fixed as zero ⇒ 1 way.
Ten's digit: any remaining digit from 0–9 except the three used (unit, thousand, zero) ⇒ 10 – 3 = 7 choices.
Count = 5 × 8 × 1 × 7 = 280. (The solution wrote 7×1×8×5 but same product).
Case C: Ten's digit is zero. (Here the zero is in the tens place, not hundreds; thousand and unit non‑zero).
Unit: 5 choices.
Thousand: 8 choices (1–9 minus unit).
Ten's: fixed as zero ⇒ 1 way.
Hundred's: any remaining digit except unit, thousand, zero ⇒ 10 – 3 = 7 choices.
Count = 5 × 8 × 1 × 7 = 280.
Total = 1680 + 280 + 280 = 2240.
Answer:2240 (Option C).