- The binomial theorem expresses the expansion of \((p+q)^n\) using coefficients \(\binom{n}{k}\).
- For \((p+q)^6\), we compute the coefficients for \(k = 0, \dots, 6\) using the formula \(\binom{6}{k} = \frac{6!}{k!(6-k)!}\).
- The resulting sequence of coefficients is \(1, 6, 15, 20, 15, 6, 1\).
- This sequence corresponds to option \(\boxed{d}\).
Explanation
- Understanding the Binomial Theorem
The expansion of a binomial power of the form \((p+q)^n\) is governed by the binomial theorem. According to this theorem, the expression can be expanded as follows:
\[(p+q)^n = \sum_{k=0}^{n} \binom{n}{k} p^{n-k} q^k\]
In our specific case, \(n = 6\). The coefficients in this expansion are the binomial coefficients, denoted by \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), which represent the number of ways to choose \(k\) elements from a set of \(n\) elements. To find the coefficients for \((p+q)^6\), we need to compute \(\binom{6}{k}\) for \(k = 0, 1, 2, 3, 4, 5, 6\).
- Calculating the Coefficients
We calculate each binomial coefficient individually:
- For \(k=0\): \(\binom{6}{0} = \frac{6!}{0!6!} = 1\)
- For \(k=1\): \(\binom{6}{1} = \frac{6!}{1!5!} = 6\)
- For \(k=2\): \(\binom{6}{2} = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15\)
- For \(k=3\): \(\binom{6}{3} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\)
- For \(k=4\): \(\binom{6}{4} = \binom{6}{2} = 15\)
- For \(k=5\): \(\binom{6}{5} = \binom{6}{1} = 6\)
- For \(k=6\): \(\binom{6}{6} = \binom{6}{0} = 1\)
Due to the symmetry property of binomial coefficients, \(\binom{n}{k} = \binom{n}{n-k}\), we observe that the sequence of coefficients is symmetric: \(1, 6, 15, 20, 15, 6, 1\).
- Identifying the Correct Answer
Comparing the calculated sequence \(1, 6, 15, 20, 15, 6, 1\) with the provided options:
A\(1, 8, 28, 56, 70, 56, 28, 8, 1\)
B\(1, 5, 10, 10, 5, 1\)
C\(1, 4, 6, 4, 1\)
D\(1, 6, 15, 20, 15, 6, 1\)
The sequence matches option d. Thus, the coefficients for the binomial expansion of \((p+q)^6\) are \(1, 6, 15, 20, 15, 6, 1\). The correct choice is \(\boxed{d}\).
Examples
Binomial coefficients are fundamental in probability theory and combinatorics. For instance, if you are flipping a coin 6 times, the binomial coefficients help you determine the number of ways to get exactly \(k\) heads. Specifically, the coefficient \(\binom{6}{k}\) tells you there are exactly 20 ways to get 3 heads in 6 coin tosses. This knowledge allows for calculating the probability of specific outcomes in any scenario with two possible results, such as success or failure in experiments, or selecting items from a limited collection.