ITPEC FE Morning April 2019 Question 2
How many four-digit decimal numbers are made from four (4) different digits between 0 and 9? Here, a four-digit decimal number has a non-zero leading digit.
Explanation
4536 — fill the leading digit first, because it has one restriction the other positions do not: it cannot be 0.
- •1st digit: 1–9, so 9 choices
- •2nd digit: any of 0–9 except the one used, so 9 choices
- •3rd digit: 8 choices
- •4th digit: 7 choices
Total: 9 × 9 × 8 × 7 = 4536.
Check by subtraction: all arrangements of 4 different digits are 10 × 9 × 8 × 7 = 5040; those starting with 0 are 9 × 8 × 7 = 504; 5040 − 504 = 4536.
Answer (c)
Why not others:
- •(d) 5040 — counts every ordered choice of 4 different digits, including the 504 that start with 0 and are really three-digit numbers
- •(a) 4320 — subtracts 10 × 9 × 8 = 720 for the leading-zero cases, but once 0 is placed only 9 digits remain for the other three positions, so there are 504 such cases
- •(b) 4436 — exactly 100 below the correct count; no consistent way of counting produces it
Key rule: In counting problems, place the most restricted position first; alternatively count everything and subtract the forbidden cases, counting those with the restriction already applied.
Basic Theory · AI-written, may contain errors
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