Question

(b) The value of the constant term in the trinomial $\left(x^2+\frac{1}{x^2}-2\right)^{10}$ is also equal to

• A. number of different dissimiliar terms in $\left( x _1+ x _2+\ldots \ldots \ldots+ x _{10}\right)^{10}$.
• B. $\left({ }^{10} C _0\right)^2+\left({ }^{10} C _1\right)^2+\left({ }^{10} C _2\right)^2+\ldots \ldots . .+\left({ }^{10} C _{10}\right)^2$
• C. coefficient of $x ^{10}$ in $(1- x )^{20}$.
• D. number of linear arrangements of 20 things of which 10 alike of one kind and rest all are different, taken all at a times.

Answer 1

Answer: (B), (C)

$\left(x^2+\frac{1}{x^2}-2\right)^{10}=\left(x-\frac{1}{x}\right)^{20}$
$T _{ r +1}={ }^{20} C _{ r } x ^{20- r }(-1)^{ r } \frac{1}{ x ^{ r }}={ }^{20} C _{ r } x ^{20-2 r }(-1)^{ r } $
$\therefore 20-2 r =0 \Rightarrow r =10$
So, term independent of $x ={ }^{20} C _{10}$. Now verify alternatives