The number of ways to choose r objects from n types, allowing repetition: nHr = (n+r−1)Cr
Example①
Choosing 5 fruits from 3 types (repetition allowed): 3H5 = 7C5 = 7C2 = 21 ways
Choosing 5 fruits from 3 types (repetition allowed): 3H5 = 7C5 = 7C2 = 21 ways
Example② (Integer Solutions of an Equation)
Nonnegative integer solutions of x+y+z=7: 3H7 = 9C7 = 36
Nonnegative integer solutions of x+y+z=7: 3H7 = 9C7 = 36
Key Points
- Combinations with repetition → use separators (○ and |)
- Positive integer solutions → transform the variables and reduce to nonnegative integer solutions
Practice Problems
- How many ways can 3 sweets be chosen from 4 types (repetition allowed)?
- Find the number of nonnegative integer solutions of x+y+z=10.
- Find the number of integer solutions of x+y+z=8 (x≥1, y≥2, z≥0).
Answers
- 4H3=6C3=20 ways
- 3H10=12C2=66
- With x'=x−1, y'=y−2, x'+y'+z=5 → 3H5=21