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Question: <sup>10</sup>C<sub>1</sub> (x – 1)<sup>2</sup> – <sup>10</sup>C<sub>2</sub> (x – 2)<sup>2</sup> + <s...

10C1 (x – 1)2 – 10C2 (x – 2)2 + 10C3 (x – 3)2 + ……. –10C10

(x – 10)10 is equal to

A

(x2 – 5) (x – 1)

B

x2 – 1

C

x2 + 1

D

x2

Answer

x2

Explanation

Solution

S = 10C1(x – 1)2 – 10C2 (x – 2)2 + 10C3(x – 3)2

+ ………10C10 (x – 10)10

S = ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}}10Cr (x – r)2

= ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}}10Cr (x2 – 2x r + r2)

= ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}}10Cr (x2) – 2x ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}}10Cr r

+ ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}} 10Cr r2

= x2 ∑r=110(–1)r+1\sum_{r = 1}^{10}{(–1)^{r + 1}}10Cr – 2x (0)

= x2 (10C1 – 10C2 + 10C3 – 10C4 + ……….. – 10C10)

= x2 (1) = x2