TheCATExam
Sign in

CAT 2022 Slot 2 — QA Question 2

Numbers (P&C)Easy

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

Answer & solution

  • A

    1200

  • B

    1420

  • 1440

  • D

    1480

Solution

Easy

Digits available: 0,1,2,3,4,50,1,2,3,4,5, each used at most once. "Greater than 20002000" means the number has 44, 55 or 66 digits (any 44-digit number ≥2000\ge 2000 qualifies, every 55- and 66-digit number automatically exceeds 20002000). Count each length by filling positions, never letting the leading digit be 00.

1

4-digit numbers >2000>2000. The leading digit must be ≥2\ge 2, so it is one of {2,3,4,5}\{2,3,4,5\} — 44 choices. The remaining three places are filled from the leftover 55 digits:

4⏟1000s×5⏟100s×4⏟10s×3⏟1s=240\underbrace{4}_{\text{1000s}}\times\underbrace{5}_{\text{100s}}\times\underbrace{4}_{\text{10s}}\times\underbrace{3}_{\text{1s}}=240
2

5-digit numbers. Leading digit just must be non-zero: 55 choices (11–55); fill the rest from the remaining 55 digits:

5×5×4×3×2=6005\times5\times4\times3\times2=600
3

6-digit numbers. Use all six digits; leading digit non-zero gives 55 choices, then arrange the remaining five:

5×5!=5×120=6005\times5!=5\times120=600

240+600+600=1440240+600+600=\mathbf{1440} — option (c).

CAT 2022 Slot 2 QA Q2: The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit — Solution | TheCATExam