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Binomial Theorem question

2002 · Shift 0 · Q105

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JEE Main
Mathematics
Binomial Theorem
2002 · Shift 0 · Q105

Binomial Theorem question

2002 · Shift 0 · Q105

JEE MainMathematicsBinomial TheoremMCQ+4 / −1

The coefficients of

{x^p}

and

{x^q}

in the expansion of

{\left( {1 + x} \right)^{p + q}}

are

A
equal
B
equal with opposite signs
C
reciprocals of each other
D
none of these

View written solutionFree

Correct answer: A

Write the general term in the expansion

For $(1+x)^{p+q},$ the coefficient of $x^r$ is $\binom{p+q}{r}.$

So:

Coefficient of $x^p$ is $\binom{p+q}{p}.$
Coefficient of $x^q$ is $\binom{p+q}{q}.$

Use the symmetry property of binomial coefficients

We know that $\binom{n}{r}=\binom{n}{n-r}.$

Here, taking $n=p+q$ : $\binom{p+q}{p}=\binom{p+q}{(p+q)-p}=\binom{p+q}{q}.$

Thus, the coefficients of $x^p$ and $x^q$ are equal.

Check the options

A: equal — Correct
B: equal with opposite signs — Incorrect, because all coefficients in $(1+x)^{p+q}$ are positive
C: reciprocals of each other — Incorrect
D: none of these — Incorrect

Therefore, the correct option is: $\boxed{A}$

Comparison with stored correct answer

Stored correct answer: A

My derived answer is also A, so they agree.

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