Question

The number of distinct terms in the expansion of ${(x+\frac{1}{x}+x^2+\frac{1}{x^2})}^{13}$ is/are

• A. $53$
• B. $61$
• C. $127$
• D. None of these

Answer 1

The correct option is A $53$

Taking $\frac{1}{x^2}$ common we get, $\frac{1}{x^{26}}(x^3+x+x^4+1{)}^{13}\to \frac{1}{x^{26}}(x^3(x+1)+x+1{)}^{13}$

$=\frac{1}{x^{26}}\left(\right(x^3+1\left)\right(x+1){)}^{13}\to \frac{1}{x^{26}}(x^3+1{)}^{13}(x+1{)}^{13}$

$=\frac{1}{x^{26}}{\sum }_{i=0}^{13}(\frac{13}{i})x^{3i}{\sum }_{j=0}^{13}(\frac{13}{j})x^j\to ={\sum }_{i=0}^{13}(\frac{13}{i}){\sum }_{j=0}^{13}(\frac{13}{j})x^{3i+j-26}$

Number of elements in the set:

$\{3i+j-26:i,jϵ\{0,1,2,.....,13\left\}\right\}=\{-26,-25,....,0,1,....25,26\}$

So, total distinct terms will be $53$.