Find the coefficient of $x^{4}$ in the expansion of $(1 + x)^{n} (1 - x)^{n}$ . Deduce that $C_{2} = C_{0} C_{4} - C_{1} C_{3} + C_{2} C_{2} - C_{3} C_{1} + C_{4} C_{0},$ where, $C$ stands for $^{n} C_{r} .$

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Solution

$(1 + x)^{n} \times (1 - x)^{n} = [C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}] \times [C_{0} - C_{1} x + C_{2} x^{2} - . . . . + (- 1)^{n} C_{n} x^{n}]$

$(1 + x)^{n} \times (1 - x)^{n} = (1 - x^{2})^{n} = [C_{0} - C_{1} x^{2} + C_{2} x^{4} - . . . + (- 1)^{n} C_{n} x^{2 n}] .$

$∴ C_{0} C_{4} - C_{1} C_{3} + C_{2} C_{2} - C_{3} C_{1} + C_{4} C_{0} = C_{2}$

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