If f(x)= xC0 xC1 x+1C12xC12xC22(x+1)C26xC26xC36(x+1)C3, then f(200) is

xC0 xC1 x+1C12xC12xC22(x+1)C26xC26xC36(x+1)C31xx+12xx(x−1)(x+1)(x)3x(x−1)x(x−1)(x−2)(x+1)x(x−1)C3→C3−C1+C2⇒1x02xx(x−1)03x(x−1)x(x−1)(x−1)0=0For any value of x → f(x) = 0.

If f(x)= xC0 xC1 x+1C12xC12xC22(x+1)C26xC26xC36(x+1)C3, then f(200) is

xC0 xC1 x+1C12xC12xC22(x+1)C26xC26xC36(x+1)C31xx+12xx(x−1)(x+1)(x)3x(x−1)x(x−1)(x−2)(x+1)x(x−1)C3→C3−C1+C2⇒1x02xx(x−1)03x(x−1)x(x−1)(x−1)0=0For any value of x → f(x) = 0.

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If $f (x) = |\begin{array}{ccc} ^{x} C_{0} & ^{x} C_{1} & ^{x + 1} C_{1} \\ 2^{x} C_{1} & 2^{x} C_{2} & 2^{(x + 1)} C_{2} \\ 6^{x} C_{2} & 6^{x} C_{3} & 6^{(x + 1)} C_{3} \end{array}|$ , then f(200) is

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} ^{x} C_{0} & ^{x} C_{1} & ^{x + 1} C_{1} \\ 2^{x} C_{1} & 2^{x} C_{2} & 2^{(x + 1)} C_{2} \\ 6^{x} C_{2} & 6^{x} C_{3} & 6^{(x + 1)} C_{3} \end{array}| \\ |\begin{array}{ccc} 1 & x & x + 1 \\ 2 x & x (x - 1) & (x + 1) (x) \\ 3 x (x - 1) & x (x - 1) (x - 2) & (x + 1) x (x - 1) \end{array}| \end{array}$

$\begin{array}{l} C_{3} \to C_{3} - (C_{1} + C_{2}) \\ \Rightarrow |\begin{array}{ccc} 1 & x & 0 \\ 2 x & x (x - 1) & 0 \\ 3 x (x - 1) & x (x - 1) (x - 1) & 0 \end{array}| = 0 \end{array}$