Combinations with Repetition?

更新：
題目問 「How many "positive integer solutions" are there?」
即 x₁, x₂, x₃ ≠ 0 嘅解法 ?

如果係嘅話
Method 1
x₁ + x₂ + x₃ = 7, x₁, x₂, x₃ ≠ 0
_, _, _, _, _, _, _, |, | 嘅排法 ( 但 |, | 唔可以排埋一齊 )
∴ C(6,2) = 15

Method 2
y₁ + y₂ + y₃ = 4, y₁, y₂, y₃ ≥ 0
[ x₁ = y₁ + 1, x₂ = y₂ + 1, x₃ = y₃ + 1 ]
H(3,4) = C(4+3-1,4) = C(4+3-1,3-1) = 15

∴ 15


若 x₁, x₂, x₃ 可以 = 0
x₁ + x₂ + x₃ = 7
Method 1：
1.
_ _ | | _ _ _ _ _：x₁ = 2, x₂ = 0, x₃ = 5
2.
| _ _ _ _ | _ _ _：x₁ = 0, x₂ = 4, x₃ = 3
......
∴ We need to permute _, _, _, _, _, _, _, |, |
But _, _, _, _, _, _, _ are the same
Also, |, | are the same

∴ P(9,2) P(7,7) / (2!7!) = P(9,7) P(2,2) / (2!7!) = 36
[ C(9,2) = C(9,7) = 36 ]


Method 2：
H(3,7) = C(7+3-1,7) = C(7+3-1,3-1) = 36

∴ 36