The ratio of the coefficient of the middle term in the expansion of...

The ratio of the coefficient of the middle term in the expansion of $( 1 + x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $( 1 + x{)}^{19}$ is .

Correct answer

Step-by-step solution

{\left(1 + x\right)}^{n}

, if

n

is even, then the middle term is

{\left(\dfrac{n}{2} + 1\right)}^{th}

term.

So, Coefficient of middle term in $( 1 + x{)}^{20} = C10 20$

{\left(1 + x\right)}^{n}

, if

n

is odd, then the middle term is

{\left(\dfrac{n + 1}{2}\right)}^{th} \text{ and } {\left(\dfrac{n + 3}{2}\right)}^{th}

term.

So, Sum of Coefficient of two middle terms in

$( 1 + x{)}^{19} = C9 19 + C10 19$

So required ratio $= \dfrac{C10 20}{C9 19 + C10 19} = \dfrac{C10 20}{C10 20} = 1$

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