Worked solution
Count numbers by their number of digits
Given the digits 1, 2, 3, 4, 6 and 8, and using each digit at most once, determine how many numbers can be formed if the number has to be
(i) less than 100,
(ii) between 100 and 1000,
(iii) between 1000 and 10000.
CJC · Discussion · Question 1
Answer
(i) 36; (ii) 120; (iii) 360.
(i) Separate one and two digits
Two-digit numbers have 6 choices then 5 choices, giving . One-digit numbers have 6 choices. The disjoint cases give .
(ii) Three digits
The source choice boxes have 6,5,4 choices in order. Thus . No digit is zero, so the leading position has no special exclusion.
(iii) Four digits
The choice boxes have 6,5,4,3 choices in order, giving .
Common mistakes
Order matters for numbers; choosing a set of digits alone is insufficient.
Solution notes
All three source cases and every choice-box count retained in text. The ordinary positive-integer interpretation excludes empty strings and leading-zero issues.