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  4. The number of terms in the expansion of (5(1/6) + 7(1/9))1824 which are integers is

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Q. The number of terms in the expansion of $\left(5^{\frac{1}{6}} + 7^{\frac{1}{9}}\right)^{1824}$ which are integers is

NTA AbhyasNTA Abhyas 2020Binomial Theorem

Solution:

$T_{K + 1}=^{1824}C_{K} \, 5^{\frac{1824 - K}{6}} \, 7^{\frac{K}{9}}$
Where, $K=0,1,2.....,1824$
Now, $\frac{1824 - K}{6}$ will be an integer if
$K=0,6,12,18,....,1824$
Also, $\frac{K}{9}$ will be an integer if $K=0,9,18,.....,1818.$
$\Rightarrow $ Both are integers if $K=0,18,36,.....,1818$
If the number of terms $=n,$
$\Rightarrow 1818=0+\left(n - 1\right)18\Rightarrow n=102$