{If the coefficients of } a^{m} { and } aⁿ { in the expansion of } (1 + a)^{m + n} { are } α { and } β {, then which one of the following is correct?}

Correct Answer is : 2α=β2α = β2α=β

Correct Answer is : 2α=β2α = β2α=β

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UPSC NDA I 2018 Mathematics Solved Paper

Question : 38 of 120

Marks: +1, -0

\text{If the coefficients of } a^{m} \text{ and } a^{n} \text{ in the expansion of } (1 + a)^{m + n} \text{ are } \alpha \text{ and } \beta \text{, then which one of the following is correct?}

Solution:

Concept:

In the binomial expansion of

\left(1+a\right)^{m+n}

, the general term is

T_{r+1} = \binom{m+n}{r} a^r

Explanation:

The coefficient of

a^m

occurs when

r = m

, so

\alpha = \binom{m+n}{m}

The coefficient of

a^n

occurs when

r = n

, so

\beta = \binom{m+n}{n}

Using the symmetry property

\binom{n}{r} = \binom{n}{n-r}

, we rewrite

\binom{m+n}{n} = \binom{m+n}{(m+n)-n} = \binom{m+n}{m}

Therefore

\beta = \binom{m+n}{m} = \alpha

Answer:

\alpha = 2\beta

, which corresponds to option B.

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