- Apply the Binomial Theorem \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) with \(n=6\).
- Calculate each term for \(k=0, 1, 2, 3, 4, 5, 6\) to get \({ }_6C_0 a^6 + { }_6C_1 a^5 b + { }_6C_2 a^4 b^2 + { }_6C_3 a^3 b^3 + { }_6C_4 a^2 b^4 + { }_6C_5 a b^5 + { }_6C_6 b^6\).
- Compare the derived expansion to the provided options.
- Conclude that the correct expansion is \(\boxed{c}\).
Explanation
- Understand the Binomial Theorem
The binomial theorem provides a systematic way to expand powers of a binomial expression. For any non-negative integer \(n\), the expression \((a + b)^n\) can be expanded as:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
where \(\binom{n}{k}\) is the binomial coefficient, often denoted as \({}_nC_k\), which is equal to \(\frac{n!}{k!(n-k)!}\). In this problem, we are asked to expand \((a + b)^6\), so we set \(n = 6\).
- Expand the Binomial Expression Term by Term
Using the formula with \(n = 6\), the expansion consists of \(n + 1 = 7\) terms. Substituting \(n = 6\) into the summation, we have:
\[(a + b)^6 = \sum_{k=0}^{6} \binom{6}{k} a^{6-k} b^k\]
Writing out the sum term by term for \(k = 0, 1, 2, 3, 4, 5, 6\):
\(\text{For } k=0: \binom{6}{0} a^{6-0} b^0 = \binom{6}{0} a^6\)
\(\text{For } k=1: \binom{6}{1} a^{6-1} b^1 = \binom{6}{1} a^5 b\)
\(\text{For } k=2: \binom{6}{2} a^{6-2} b^2 = \binom{6}{2} a^4 b^2\)
\(\text{For } k=3: \binom{6}{3} a^{6-3} b^3 = \binom{6}{3} a^3 b^3\)
\(\text{For } k=4: \binom{6}{4} a^{6-4} b^4 = \binom{6}{4} a^2 b^4\)
\(\text{For } k=5: \binom{6}{5} a^{6-5} b^5 = \binom{6}{5} a b^5\)
\(\text{For } k=6: \binom{6}{6} a^{6-6} b^6 = \binom{6}{6} b^6\)
- Compare with Given Options and Select the Correct Answer
Summing these terms together, we get the complete expansion:
\[(a + b)^6 = \binom{6}{0} a^6 + \binom{6}{1} a^5 b + \binom{6}{2} a^4 b^2 + \binom{6}{3} a^3 b^3 + \binom{6}{4} a^2 b^4 + \binom{6}{5} a b^5 + \binom{6}{6} b^6\]
Comparing this to the given options:
Option a is incomplete as it misses the final term \({}_6C_6 b^6\).
Option b is an alternative representation but option c follows the standard ascending order of powers of \(b\).
Option c matches our derived result: \({ }_6C_0 a^6 + { }_6C_1 a^5 b + { }_6C_2 a^4 b^2 + { }_6C_3 a^3 b^3 + { }_6C_4 a^2 b^4 + { }_6C_5 a b^5 + { }_6C_6 b^6\).
Option d has incorrect powers of \(a\) and \(b\).
Therefore, the correct choice is option c.
Examples
The binomial theorem is widely used in probability and statistics, particularly in modeling scenarios where there are two possible outcomes (success or failure) over a fixed number of trials. For example, if you flip a coin 6 times, the probability of getting exactly \(k\) heads can be calculated using the binomial distribution formula, which is directly derived from the coefficients in the expansion of \((a + b)^6\). This makes the theorem a powerful tool for analyzing games of chance, quality control in manufacturing, and even predicting patient recovery rates in medical studies.