Which of the following statement(s) is/are true?

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The value of ${}^{404}C_{4} - {}^{4}C_{1} {}^{303}C_{4} + {}^{4}C_{2} {}^{202}C_{4} - {}^{4}C_{3} {}^{101}C_{4}$ is equal to (404)⁴.

Let ${(1 + x + x^{2})}^{25} = a_{0} + a_{1} x + a_{2} x^{2} + ........ + a_{50} x^{50}$ then the value of $a_{3} - a_{5} + a_{7} - a_{9} + ........... - a_{49} = 24$

The value of $S = \sum_{r = 0}^{11} \frac{{(- 1)}^{r} ({}^{11}C_{r}) 2^{r + 1}}{(r + 1) (r + 2)}$ is equal to $\frac{1}{13}$

The constant term in the expansion of ${(x + x^{2} + \frac{1}{x} + \frac{1}{x^{2}})}^{15}$ is $\sum_{r = 0}^{5} ({}^{15}C_{r + 5}) ({}^{15}C_{3 r})$

answer is B, C, D.

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Detailed Solution

(A) Co-efficient of x⁴ in $[{}^{4}C_{0} {(1 + x)}^{404} - {}^{4}C_{1} {(1 + x)}^{303} + .............]$

= Co-efficient of x4 in ${({(1 + x)}^{101} - 1)}^{4} \equiv {(101)}^{4}$

(B) $a_{1} - a_{3} + a_{5} - a_{7} + ........... + a_{49} = Im {(1 + i + i^{2})}^{25} = 1$

$\Rightarrow a_{3} + a_{5} - a_{7} - .......... = a_{1} - 1 = 25 - 1 = 24$

a1 can be found by differentiating the series and putting x = 0.
(C) $S = \sum_{r = 0}^{11} \frac{{(- 1)}^{r} {}^{11}C_{r} 2^{r + 1}}{(r + 1) (r + 2)} = \frac{1}{2 \times 12 \times 13} \sum_{r = 0}^{11} {}^{13}C_{r + 2} {(- 2)}^{r + 2}$

(D) ${(x + x^{2} + \frac{1}{x} + \frac{1}{x^{2}})}^{15} = {(x + x^{2})}^{15} {(1 + \frac{1}{x^{3}})}^{15}$
$= x^{15} {{}^{15}C_{0} + {}^{15}C_{1} x + ... {}^{15}C_{15} x^{15}} {{}^{15}C_{0} + {}^{15}C_{1} \frac{1}{x^{3}} + {}^{15}C_{2} \frac{1}{x^{6}} + ... {}^{15}C_{15} \frac{1}{x^{45}}}$

Constant terms add up to
${}^{15}C_{5} {}^{15}C_{0} + {}^{15}C_{6} {}^{15}C_{3} + {}^{15}C_{7} {}^{15}C_{6} + {}^{15}C_{8} {}^{15}C_{9} + {}^{15}C_{9} {}^{15}C_{12} + {}^{15}C_{10} {}^{15}C_{15}$