Verified Answer :

  • The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
  • Using \(n=3\), the coefficients are calculated as \(\binom{3}{0}=1\), \(\binom{3}{1}=3\), \(\binom{3}{2}=3\), and \(\binom{3}{3}=1\).
  • The resulting coefficients for the expansion \((a+b)^3\) are \(1, 3, 3, 1\).
  • This corresponds to option d, which is \(\boxed{d}\).

Explanation

  1. The Binomial Theorem

The binomial expansion of \((a+b)^n\) is given by the formula:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
where \(\binom{n}{k}\) represents the binomial coefficient calculated as:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
In our problem, we have \(n = 3\). The goal is to identify the coefficients of this expansion, which are \(\binom{3}{0}, \binom{3}{1}, \binom{3}{2},\) and \(\binom{3}{3}\).

  1. Calculating the Binomial Coefficients

To find the coefficients for \((a+b)^3\), we evaluate each binomial coefficient individually:

  • For \(k = 0\): \(\binom{3}{0} = \frac{3!}{0!3!} = \frac{3 \times 2 \times 1}{1 \times (3 \times 2 \times 1)} = 1\)
  • For \(k = 1\): \(\binom{3}{1} = \frac{3!}{1!2!} = \frac{3 \times 2 \times 1}{1 \times (2 \times 1)} = 3\)
  • For \(k = 2\): \(\binom{3}{2} = \frac{3!}{2!1!} = \frac{3 \times 2 \times 1}{(2 \times 1) \times 1} = 3\)
  • For \(k = 3\): \(\binom{3}{3} = \frac{3!}{3!0!} = \frac{3 \times 2 \times 1}{(3 \times 2 \times 1) \times 1} = 1\)

Thus, the sequence of coefficients is \(1, 3, 3, 1\).

  1. Final Answer Conclusion

The expansion of \((a+b)^3\) is:
\[(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3\]
The coefficients are \(1, 3, 3, 1\). Comparing this result with the provided options:

A\(1, 2, 1\)
B\(1, 1\)
C\(1, 4, 6, 4, 1\)
D\(1, 3, 3, 1\)

The correct option is d. The final answer is \(\boxed{d}\).

Examples

Binomial coefficients are used in probability and statistics to determine the number of ways to choose items from a set, such as calculating the number of possible outcomes when flipping a coin multiple times. For example, if you flip a coin 3 times, the coefficients \(1, 3, 3, 1\) represent the frequency of getting 0, 1, 2, or 3 heads, respectively. Understanding these patterns helps in predicting outcomes in various real-world scenarios involving trials and combinations.

Related Questions