Question

The total number of terms in the expansion of $(x+a{)}^{47}-(x-a{)}^{47}$ after simplification is

• A. 24
• B. 47
• C. 48
• D. 96

Answer 1

Answer: (A)

24

${(x+a)}^{47}-{(x-a)}^{47}$

When we expand the above equation using binomial expansion

$(x+y{)}^n=\left(\sum _{k=0}^n{}^n{C}_k{x}^k{y}^{n-k}\right)$

So the above equation becomes

$(x+a{)}^{47}=\left(\sum _{k=0}^{47}{}^{47}{C}_k{x}^k{a}^{47-k}\right)$

$(x-a{)}^{47}=\left(\sum _{k=0}^{47}{}^{47}{C}_k{x}^k\right(-a{)}^{47-k})$
${(x+a)}^{47}\Rightarrow$There are 48 terms in the expansion and all are positive
${(x-a)}^{47}\Rightarrow$There are 48 terms in the expansion

The terms with odd powers of a will be cancelled and those with even powers of a will add up.
24 terms will be positive and 24 negative in the expansion of $(x-a{)}^{47}$
48 terms positive-[24 terms negative and 24 terms positive]
$=48termspositive+24termsnegative+24termspositive$
$=24$ terms