Question

In any binomial expansion, the number of terms are
A. $ \geqslant 5$
B. $ \geqslant 2$
C. $ \geqslant 3$
D. $ \geqslant 4$

Answer 1

Hint: To understand this question we need to understand the proper definition of binomial expansion to figure out what the minimum number of terms required is. We can put $n = 1,2,3...$ also if required to further check if our theory is correct in this. We need to expand the binomial expansion any further since this will be enough to prove the viability of our answer.

Formula used: The formula used in this question is the formula for the binomial expansion of two terms raised to a particular power$n$ in a way that one of the terms is $1$ and the other term is a variable $x$
The expansion can be mathematically represented as
${\left( {1 + x} \right)^n} = \sum {^n{C_r}{x^r}} $
Where $^n{C_r}{x^r}$ is the ${\left( {r + 1} \right)^{th}}$ term

Complete step-by-step solution:
The binomial expansion or theorem is a way of expanding any expression that has been raised to a finite power. This binomial theorem is a powerful tool of expansion, which has applications in multiple facets of mathematics.
As huge powers are harder to expand or calculate in the regular sense, binomial theorem is used to calculate such expressions for ease and reduction of calculation time.
In ${\left( {1 + x} \right)^n} = \sum {^n{C_r}{x^r}} $ , the lowest possible value for $n = 1$ as anything raised to power $0$ is not a binomial expansion which means the
With $n = 1$ , the total terms formed are $2$ . As we keep expanding the terms with higher powers, we find that the total number of terms possible is $n + 1$ . Thus the minimum number of terms is $2$

That means the correct answer is $\left( b \right) \geqslant 2$

Note: It isn’t necessary but if needed, you can check for higher powers to check how many terms are possible. It will always be greater than $2$ . This is simply the lowest number of terms possible.