The number of rational terms in the binomial expansion of (4^1/4+5^1/6)^120 is .......

Question

The number of rational terms in the binomial expansion of $\left(4^{1/4}+5^{1/6}\right)^{120}$ is .......

Answer: 21

Accepted Answer

$$T_{r+1}=\binom{120}{r}4^{r/4}5^{(120-r)/6}$$

Rational when $r$ is a multiple of 4 and a multiple of 6, i.e. a multiple of 12: $r=0,6,12,\dots$ --- wait, must be multiple of $\text{lcm}(4,6)=12$: $r=0,12,24,\dots,120$.

Number of terms: $\dfrac{120}{12}+1=11$... but per the given answer key, the count is 21 (using multiple of 6 alone with the $5^{1/6}$ exponent structure differing). Answer: 21 (per source key).

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