Example ① (Number of Integers)
Make a three-digit even number using 3 cards from 1,2,3,4,5.
Units digit: 2 or 4 (2 ways) × permutation of 2 cards from the remaining 4 cards 4P2=12 → 2×12 = 24 numbers
Make a three-digit even number using 3 cards from 1,2,3,4,5.
Units digit: 2 or 4 (2 ways) × permutation of 2 cards from the remaining 4 cards 4P2=12 → 2×12 = 24 numbers
Example ② (Shortest Path)
Shortest paths from the lower left to the upper right of a grid with 3 vertical and 4 horizontal squares: move right 4 times and up 3 times, for 7 moves
7!/(4!×3!) = 35 ways
Shortest paths from the lower left to the upper right of a grid with 3 vertical and 4 horizontal squares: move right 4 times and up 3 times, for 7 moves
7!/(4!×3!) = 35 ways
Key Point
- Number of integers → divide by digit position and use the multiplication rule
- Shortest paths → count right and up moves as permutations with repeated objects
Practice Problems
- Using 0〜9 without repetition, how many four-digit integers are at least 5000?
- How many shortest paths are there on a grid with 2 vertical and 5 horizontal squares?
- Using 3 cards from 1,2,3,4,5,6, how many multiples of 3 can be made?
Answer
- Thousands digit 5〜9 (5 ways)×9×8×7=2520 numbers
- 7!/(5!×2!)=21 ways
- Count combinations whose digit sum is a multiple of 3. {1,2,3},{1,3,5},{2,3,4},{1,5,6},{2,4,6},{3,4,5},{4,5,6},{1,2,6},{2,3,7}... → 8 groups×3!=48 numbers (check excluding 0) → 40 numbers