The question in the image asks to determine which expression correctly represents the expansion of Newton's binomial $(a + b)^n$ using summation notation. This is commonly known as the Binomial Theorem.

The Binomial Theorem states:

$(a + b)^n = \sum_{k = 0}^{n} \binom{n}{k} a^{n-k} b^k$

where:

• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
• $a^{n-k}$ and $b^k$ are the terms that decrease and increase their powers, respectively, as $k$ increases from 0 to $n$.

Looking at the four options in the image:

1. $\sum_{k = 0}^{n} \binom{n}{k} a^{n-k} b^k$ — This is the correct representation.
2. $\sum_{k = 0}^{n} \binom{n}{k} a^k b^k$ — This expression is incorrect because both $a$ and $b$ have the same exponent, which does not follow the pattern of the Binomial Theorem.
3. $\sum_{k = 0}^{k} \binom{n}{k} b^k a^{n-k}$ — This expression is incorrect because the limits of summation are incorrect; it should be up to $n$, not $k$.
4. $\sum_{k = 0}^{n} \binom{n}{k} a^k b^{n-k}$ — This expression is incorrect because the powers of $a$ and $b$ are swapped compared to the correct expression.

Therefore, the correct answer is the first option.

Would you like more details or have any questions?

Here are 5 related questions that expand upon this information:

1. What is the general formula for the binomial coefficient $\binom{n}{k}$?
2. How can the Binomial Theorem be applied to find specific terms in a binomial expansion?
3. Can the Binomial Theorem be extended to work for negative or fractional exponents? If so, how?
4. How can Pascal's Triangle be used to find binomial coefficients?
5. How does the Multinomial Theorem generalize the Binomial Theorem?

Tip: The Binomial Theorem is very useful in probability and combinatorics for expanding expressions and calculating coefficients in polynomials.