Combinatorics

Question

$$T_k = \,^{100}C_k \cdot x^{100-k} \quad \text{for} \quad k = 0, 1, 2, \ldots 100$$

Then, what will be the value of
$$\Bigl (T_0 - T_2 + T_4 - \cdots + T_{100} \Bigr )^2 + \Bigl (T_1 - T_3 + T_5 - \cdots - T_{99} \Bigr )^2$$

The answer should be in terms of $x$

Answer 1

Is it

100C0x^200+100C1x^198+100C2x^196+100C3x^194+.............................+1

Comments

brute force using small value...

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when i tried with small values i got something like 2^n .... for here it will be 2^100 . and in terms of x the equivalent expression is (X^2 + 1)^100 . But i need a proof of it.

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For proof ask to arjun sir.