Number of integral terms in the expansion of $\left\{7^{(1/2)} + 11^{(1/6)}\right\}^{824}$ is equal to ______.
Step-by-Step Solution
Key Concept: General term: $T_{r+1}=\binom{824}{r}\cdot 7^{(824-r)/2}\cdot 11^{r/6}$. For integral term, $(824-r)/2$ must be integer (i.e. $r$ even) AND $r/6$ must be integer (i.e. $r$ multiple of 6). So $r$ must be a multiple of 6 with $0\le r\le 824$.
$T_{r+1}=\binom{824}{r}\cdot 7^{(824-r)/2}\cdot 11^{r/6}$. For integral powers: $r$ is even AND $r$ is divisible by 6, so $r$ must be a multiple of 6. $r=0,6,12,\ldots,822$. Number of terms $=\frac{822-0}{6}+1=137+1=138$.
Correct Answer: 138