Question

Assertion :Number of the dissimilar terms in the sum of expansion $(x+a{)}^{102}+(x-a{)}^{102}$ is 206. Reason: Number of terms in the expansion of $(x+b{)}^n$ is $n+1$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

The odd terms will be the similar terms.
Hence
${}^n{C}_r$ be the binomial coefficient of the terms,
Then the terms are similar for $rϵ\{0,2,4,\mathrm{6...102}\}$
Hence no of similar terms
$\frac{102-0}{2}=n-1$
Or
$51+1=n$
Hence $n=52$.
Now total number of terms will be 103.
Hence $51$ terms will be dissimilar occur in pairs.
That adds up to $102$ dissimilar terms.
Hence in the expression $(x+a{)}^{102}+(x-a{)}^{102}$
Total number of terms that is supposed to be is equal to 206.
However even terms are dissimilar and eliminate, the number of dissimilar terms being $2\left(51\right)=102$.
Hence number of similar terms will be $206-102=104$.