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Ratio and Proportion

If a 1 and ...

Question

If $a_{1}$ and $a_{2}$ be the coefficients of $x^{4}$ and of $x^{6}$ and $(1 + x)^{10}$ respectively, then

A

a_{1} = a_{2}

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B

2 a_{1} = a_{2}

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C

a_{1} = 4 a_{2}

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D

a_{1} = 2 a_{2}

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Solution

The correct option is D $a_{1} = a_{2}$
For the above binomial expansion $(1 + x)^{10}$
$T_{r + 1} =^{10} C_{r} x^{r}$

Therefore the coefficient of $x^{4}$ will be
$^{10} C_{4} = a_{1}$

Similarly
coefficient of $x^{6}$ will be
$a_{2} =^{10} C_{6}$

Now by the property of binomial coefficient
$^{n} C_{r} =^{n} C_{n - r}$

Therefore
$^{10} C_{4} =^{10} C_{10 - 4}$
$=^{10} C_{6}$

Therefore, $a_{1} = a_{2}$

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