Forming Integers Greater Than 2000

CAT 2022 Slot 2 · QA · Medium · Modern Math

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The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit at most once, is

  1. A.

    1440

  2. B.

    1200

  3. C.

    1420

  4. D.

    1480

Answer

A

Explanation

Digits available: {0, 1, 2, 3, 4, 5} (total 6 digits). We need numbers > 2000 using digits at most once.

Case 1: 4-digit numbers > 2000 First digit can be 2, 3, 4, 5 (4 choices). Remaining 3 places chosen from remaining 5 digits: 5P3=5×4×3=60^5P_3 = 5 \times 4 \times 3 = 60. Total 4-digit numbers = 4×60=2404 \times 60 = 240.

Case 2: 5-digit numbers First digit cannot be 0 (5 choices). Remaining 4 places: 5P4=120^5P_4 = 120. Total 5-digit numbers = 5×120=6005 \times 120 = 600.

Case 3: 6-digit numbers First digit cannot be 0 (5 choices). Remaining 5 places: 5P5=120^5P_5 = 120. Total 6-digit numbers = 5×120=6005 \times 120 = 600.

Total integers = 240+600+600=1440240 + 600 + 600 = 1440.

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