- Apply the Binomial Theorem formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) with \(a=x\), \(b=3\), and \(n=12\).
- Identify the term involving \(x^4\) by setting the exponent \(12-k=4\), which gives \(k=8\).
- Substitute \(k=8\) into the general term to find \(\binom{12}{8} x^4 3^8\).
- Conclude that the coefficient is \({ }_{12}C_{8}(3)^8\), which corresponds to option b.
- Final answer: \(\boxed{b}\)
Explanation
- Understanding the Binomial Theorem
To find the coefficient of the \(x^4\)-term in the binomial expansion of \((x+3)^{12}\), we use the Binomial Theorem. The theorem states that for any non-negative integer \(n\), the expansion of \((a+b)^n\) is given by:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
where \(\binom{n}{k} = { }_{n}C_{k} = \frac{n!}{k!(n-k)!}\) is the binomial coefficient.
- Identifying the Term for \(x^4\)
In our expression \((x+3)^{12}\), we have \(a=x\), \(b=3\), and \(n=12\). The general term in this expansion is:
\[T_{k+1} = \binom{12}{k} x^{12-k} 3^k\]
To find the term containing \(x^4\), we set the exponent of \(x\) to 4:
\[12 - k = 4\]
Solving for \(k\), we get:
\[k = 12 - 4 = 8\]
So, the term corresponding to \(x^4\) occurs when \(k=8\).
- Calculating the Coefficient
Substituting \(k=8\) into the general term expression:
\[T_{8+1} = \binom{12}{8} x^{12-8} 3^8 = \binom{12}{8} x^4 3^8\]
The coefficient of the \(x^4\)-term is the constant factor multiplying \(x^4\), which is:
\[\binom{12}{8} 3^8\]
In standard notation, this is written as \({ }_{12}C_{8} (3)^8\). Comparing this to the given options, we find it matches option b.
Examples
The Binomial Theorem is a powerful tool used in probability and statistics, particularly when calculating the likelihood of specific outcomes in repeated trials, such as flipping a coin or rolling a die multiple times. For example, if you wanted to calculate the probability of getting exactly 4 heads in 12 tosses of a fair coin, you would look at the expansion of \((\frac{1}{2} + \frac{1}{2})^{12}\), which involves the same binomial coefficients used to find the expansion of \((x+3)^{12}\). Understanding this allows for efficient calculation of complex probabilities and distribution patterns.