Question 1

What is the number of terms in the expansion of $(x + y)^{n}$?

Accepted Answer

The number of terms in the expansion of $(x + y)^{n}$ is always $n + 1$, where $n$ is a positive integer. This is because the powers of $x$ decrease from $n$ to $0$ while the powers of $y$ increase from $0$ to $n$, giving a total of $(n - 0) + 1 = n + 1$ terms.

Question 2

What is the general term ($r+1$-th term) in the expansion of $(x + y)^{n}$?

Accepted Answer

The general term, denoted by $T{r+1}$, in the binomial expansion of $(x + y)^{n}$ is given by the formula $T{r+1} = {}^{n}C{r} x^{n-r} y^{r}$.

Question 3

What is the sum of binomial coefficients ${}^{n}C{0} + {}^{n}C{1} + {}^{n}C{2} + \dots + {}^{n}C{n}$?

Accepted Answer

The sum of all binomial coefficients in the expansion of $(1 + x)^{n}$ is obtained by putting $x = 1$, which yields $(1 + 1)^{n} = 2^{n}$.

Question 4

If $n$ is even, the middle term in the expansion of $(x + y)^{n}$ is:

Accepted Answer

When $n$ is an even integer, the total number of terms in $(x + y)^{n}$ is $n + 1$ (which is odd). Therefore, there is only one middle term, which is given by $T{\frac{n}{2} + 1} = {}^{n}C{\frac{n}{2}} x^{\frac{n}{2}} y^{\frac{n}{2}}$.

Question 5

If $n$ is odd, the middle terms in the expansion of $(x + y)^{n}$ are:

Accepted Answer

When $n$ is odd, the total number of terms is $n + 1$ (which is even). Consequently, there are two middle terms, given by $T{\frac{n+1}{2}}$ and $T{\frac{n+3}{2}}$.

Binomial Theorem