- The binomial theorem states \((a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\).
- Substitute \(a=m\), \(b=3\), and \(n=4\) into the theorem: \((m+3)^4 = \sum_{k=0}^4 \binom{4}{k} m^{4-k} 3^k\).
- Expanding the sum yields \({ }_4C_0 m^4 3^0 + { }_4C_1 m^3 3^1 + { }_4C_2 m^2 3^2 + { }_4C_3 m^1 3^3 + { }_4C_4 m^0 3^4\).
- This matches the first option provided. Final answer is \(\boxed{1}\).
Explanation
- Applying the Binomial Theorem
To determine which expression is equivalent to \((m+3)^4\), 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 \(\binom{n}{k}\) (often written as \({ }_nC_k\)) is the binomial coefficient, which represents the number of ways to choose \(k\) elements from a set of \(n\) elements. In our specific case, we have \(a=m\), \(b=3\), and \(n=4\).
- Expanding the Expression
By substituting \(a=m\), \(b=3\), and \(n=4\) into the general binomial expansion formula, we get:
\[(m+3)^4 = \sum_{k=0}^4 \binom{4}{k} m^{4-k} 3^k\]
Now, we expand this summation by iterating \(k\) from 0 to 4:
- For \(k=0\): \(\binom{4}{0} m^{4-0} 3^0 = { }_4C_0 m^4 3^0\)
- For \(k=1\): \(\binom{4}{1} m^{4-1} 3^1 = { }_4C_1 m^3 3^1\)
- For \(k=2\): \(\binom{4}{2} m^{4-2} 3^2 = { }_4C_2 m^2 3^2\)
- For \(k=3\): \(\binom{4}{3} m^{4-3} 3^3 = { }_4C_3 m^1 3^3\)
- For \(k=4\): \(\binom{4}{4} m^{4-4} 3^4 = { }_4C_4 m^0 3^4\)
Combining these, we obtain the full expression:
\[{ }_4C_0 m^4 3^0 + { }_4C_1 m^3 3^1 + { }_4C_2 m^2 3^2 + { }_4C_3 m^1 3^3 + { }_4C_4 m^0 3^4\]
- Conclusion and Selection
Comparing our derived expression, \({ }_4C_0 m^4 3^0 + { }_4C_1 m^3 3^1 + { }_4C_2 m^2 3^2 + { }_4C_3 m^1 3^3 + { }_4C_4 m^0 3^4\), with the given choices, we see that it perfectly matches choice 1. Other choices either misplace the exponents on \(m\) and \(3\) or use incorrect indices for the binomial coefficients. Thus, the equivalent expression is the one listed as choice 1. The final answer is \(\boxed{1}\).
Examples
The binomial theorem is an essential tool in probability and statistics, particularly when calculating binomial distributions. For example, if you are calculating the probability of getting exactly 2 heads in 4 coin flips, you look at the term corresponding to \(k=2\) in the expansion of \((0.5+0.5)^4\). Understanding how to expand binomials allows you to model complex scenarios where multiple independent events occur, providing a systematic way to calculate the outcomes of various trials.