The coefficient of $x^{n}$ in the polynomial
$(x + {^{2 n + 1} C}_{0}) (x + {^{2 n + 1} C}_{1}) (x + {^{2 n + 1} C}_{2}) + . . . . (x + {^{2 n + 1} C}_{n})$

2^{n + 1}

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2^{2 n + 1} - 1

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2^{2 n}

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2^{2 n + 1} + 1

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Solution

The correct option is C $2^{2 n}$
The required coefficient will be
$^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{n}$
Now
$^{2 n + 1} C_{r} =^{2 n + 1} C_{2 n + 1 - r}$
Now
$(1 + x)^{2 n + 1} |_{x = 1} =^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{2 n}$
The middle terms will $^{2 n + 1} C_{n}$ and $^{2 n + 1} C_{n + 1}$
Hence
$2^{2 n + 1} =^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{2 n}$
$= 2 [^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{n - 1}] +^{2 n + 1} C_{n} +^{2 n + 1} C_{n + 1}$
$= 2 [^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{n}]$
Hence
$2^{2 n + 1} = 2 [^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{n}]$
$^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} +^{2 n + 1} C_{3} . . .^{2 n + 1} C_{n} = 2^{2 n}$
Thus the coefficient is $2^{n}$ .

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Similar questions

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