(b) The value of the constant term in the trinomial (x2+(1/x2)-2)10 is also equal to

Question

(b) The value of the constant term in the trinomial (x2+(1/x2)-2)10 is also equal to (A) number of different dissimiliar terms in ( x 1+ x 2+ ldots ldots ldots+ x 10)10. (B) ( 10 C 0)2+( 10 C 1)2+( 10 C 2)2+ ldots ldots . .+( 10 C 10)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.

Tardigrade Admin · Accepted Answer

(x2+(1/x2)-2)10=(x-(1/x))20 T r +1= 20 C r x 20- r (-1) r (1/ x r )= 20 C r x 20-2 r (-1) r ∴ 20-2 r =0 ⇒ r =10 So, term independent of x = 20 C 10. Now verify alternatives

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

number of different dissimiliar terms in $(x_{1} + x_{2} + \dots\dots\dots + x_{10})^{10}$ .

$(^{10} C_{0})^{2} + (^{10} C_{1})^{2} + (^{10} C_{2})^{2} + \dots\dots .. + (^{10} C_{10})^{2}$

coefficient of $x^{10}$ in $(1 - x)^{20}$ .

number of linear arrangements of 20 things of which 10 alike of one kind and rest all are different, taken all at a times.

$(x^{2} + \frac{1}{x ^{2}} - 2)^{10} = (x - \frac{1}{x})^{20}$
$T_{r + 1} =^{20} C_{r} x^{20 - r} (- 1)^{r} \frac{1}{x ^{r}} =^{20} C_{r} x^{20 - 2 r} (- 1)^{r}$
$∴ 20 - 2 r = 0 \Rightarrow r = 10$
So, term independent of $x =^{20} C_{10}$ . Now verify alternatives

Q. (b) The value of the constant term in the trinomial (x2+x21​−2)10 is also equal to

number of different dissimiliar terms in (x1​+x2​+………+x10​)10.

(10C0​)2+(10C1​)2+(10C2​)2+……..+(10C10​)2

coefficient of x10 in (1−x)20.