<s>[INST] Multinomial theorem member that does not contain an irrational number Hey I am supposed to solve the following problem:

Specify a development member that does not contain an irrational number:
  $$\left (\sqrt{5} - \sqrt[3]{2} +2 \right )^{6}.$$

So I used multinomial theorem:
$$\sum_{i=0}^{6}\binom{6}{n_{1},n_{2},n_{3}}\left ( \sqrt{5} \right )^{n_{1}}\left (  - \sqrt[3]{2} \right )^{n_{2}}\left ( 2 \right )^{n_{3}}$$
and then I know that :$ 2k+3l +n_{3}=6$ with $n_1=2k$ and $n_2=3l$.
Is that a correct answer?
$$\binom{6}{2,3,1}+\binom{6}{2,0,4}+\binom{6}{4,0,2}+\binom{6}{6,0,0}+\binom{6}{0,3,3}+\binom{6}{0,0,6}$$
 [/INST] No, your answer is not correct. Since $n_1\in\{0,2,4,6\}$ and $n_2\in\{0,3,6\}$ with $n_1+n_2\leq 6$, we find that the required number is
$$N:=\sum_{(n_1,n_2)\in S}\binom{6}{n_{1},n_{2},6-n_1-n_2}\left ( \sqrt{5} \right )^{n_{1}}\left (  - \sqrt[3]{2} \right )^{n_{2}}\left ( 2 \right )^{6-n_1-n_2}$$
where $S:=\{(0,0), (0,3), (0,6), (2,0), (2,3),(4,0), (6,0)\}$. What is $N$?
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