- Identify the given expression as the binomial expansion of \((m+3)^4\) using the Binomial Theorem formula \(\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\).
- Substitute \(x\) for \(m\) and expand \((x+3)^4\) term by term: \(x^4 + 12x^3 + 54x^2 + 108x + 81\).
- Compare the expanded result to the provided options.
- The equivalent expression is \(\boxed{a}\).
Explanation
- Identifying the Binomial Expansion
The given expression is
\[{ }_4 C_0 m^4 3^0 + { }_4 C_1 m^3 3^1 + { }_4 C_2 m^2 3^2 + { }_4 C_3 m^1 3^3 + { }_4 C_4 m^0 3^4.\]
The Binomial 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 \(\binom{n}{k} = { }_n C_k\). Comparing our given expression to the general formula, we identify \(n=4\), \(a=m\), and \(b=3\). Therefore, the expression is the binomial expansion of \((m+3)^4\).
- Expanding the Expression
Assuming \(m\) corresponds to the variable \(x\), we are looking for the expansion of \((x+3)^4\). We calculate each term:
- The first term is \(\binom{4}{0} x^4 3^0 = 1 \cdot x^4 \cdot 1 = x^4\).
- The second term is \(\binom{4}{1} x^3 3^1 = 4 \cdot x^3 \cdot 3 = 12x^3\).
- The third term is \(\binom{4}{2} x^2 3^2 = 6 \cdot x^2 \cdot 9 = 54x^2\).
- The fourth term is \(\binom{4}{3} x^1 3^3 = 4 \cdot x^1 \cdot 27 = 108x\).
- The fifth term is \(\binom{4}{4} x^0 3^4 = 1 \cdot 1 \cdot 81 = 81\).
Summing these terms, we obtain \(x^4 + 12x^3 + 54x^2 + 108x + 81\).
- Comparing with Given Options
Comparing our result \(x^4 + 12x^3 + 54x^2 + 108x + 81\) to the provided options:
A\(x^4 + 12x^3 + 54x^2 + 108x + 81\)
B\(4x^4 + 12x^3 + 54x^2 + 108x + 324\)
C\(4x^4 + 12x^3 + 54x^2 + 108x + 81\)
The expanded expression exactly matches option a. Thus, the expression is equivalent to \(x^4 + 12x^3 + 54x^2 + 108x + 81\).
Examples
In fields like probability or statistics, the binomial theorem is essential for calculating the probabilities of success in a sequence of independent trials. For example, if you are calculating the likelihood of getting a specific number of "heads" in multiple coin tosses, you use binomial expansion to find the coefficients. Understanding these patterns helps in predicting outcomes in real-world scenarios like product quality control (pass/fail tests) or clinical trial success rates, where each event has a binary outcome.