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k, n ∈ N, n ≥­ 3. How many integer solutions are there to the equation: $x_1 + · · · + x_n = k$ , such that: $∀(1≤i≤ _nx_i≥0)$ ­and x1 ≤ 100, x2 ≤ 200, x3 ≤ 300?

I tried solving by by doing something like this

$N(\bar{C_1}\bar{C_2}...\bar{C_n}) = S_0 - S_1 + ... + (-1)^n(S_n)$

$C_i$ = there's a solution $(1 ≤ i ≤ _nx_i ≥ 0)$

Where $S_0$ = there's 0 solutions and $S_n$ = there's n possible solutions

Assuming for n = 3

$x_1 + x_2 + x_3 = k$

$\bar{x_1} + \bar{x_2} + \bar{x_3} = k - 100 - 200 - 300$

$N(\bar{C_1}\bar{C_2}\bar{C_3}) = N - [N(C_1) + N(C_2) + N(C_3) + N(C_4)] + [N(C_1C_2) + N(C_1C_3) + N(C_1C_4) + N(C_2C_3) + N(C_2C_4) + N(C_3C_4)] - [N(C_1C_2C_3)]$

N = $k + 3 - 1 \choose k$

$\bar{x_1} + x_2 + x_3 = k - 100$

$S_1$ = $k - 100 + 3 - 1 \choose k - 100$$3 \choose 1$ Since there's 3 possibilities

I'm not sure if I'm going in the right direction so I stopped solving further, any help would be appreciated.

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    This can be obtained as coefficient of $x^k$ in the expansion $(1+x+\cdots+x^{100})(1+x+\cdots+x^{200})(1+x+\cdots+x^{300})(1+x+\cdots)^{n-3} = (1-x^{101})(1-x^{201})(1-x^{301})(1-x)^{-n}$ – 2017-01-17

0 Answers 0