- The binomial theorem formula for the general term of \((A+B)^n\) is \(T_{k+1} = \binom{n}{k} A^{n-k} B^k\).
- By setting \(A=e\), \(B=2f\), \(n=10\), and \(k=3\), we find the fourth term is \(T_4 = \binom{10}{3} e^7 (2f)^3\).
- Matching this with the provided choices, we identify option c as the correct expression.
- The final answer is \(\boxed{c}\).
Explanation
- Understanding the Binomial Expansion Formula
The binomial theorem states that the expansion of a binomial power \((A + B)^n\) is given by the sum:
\[(A + B)^n = \sum_{k=0}^{n} \binom{n}{k} A^{n-k} B^k\]
where \(\binom{n}{k}\) is the binomial coefficient, often written as \({}_{n}C_k\). In this problem, we are given the expression \((e + 2f)^{10}\). We identify the components as:
- \(A = e\)
- \(B = 2f\)
- \(n = 10\)
The general term of this expansion is given by \(T_{k+1} = \binom{10}{k} e^{10-k} (2f)^k\).
- Calculating the Fourth Term
To find the fourth term of the expansion, we need to set the term index to \(k+1=4\), which implies \(k=3\). Using the general formula for the term:
\[T_4 = \binom{10}{3} e^{10-3} (2f)^3\]
Substituting the values:
- \(\binom{10}{3}\) is the coefficient.
- \(e^{10-3} = e^7\).
- \((2f)^3\) is the variable part.
So, the fourth term is \(\binom{10}{3} e^7 (2f)^3\).
- Selecting the Correct Option
Comparing our result \({}_{10}C_3(e^7)(2f)^3\) with the given options:
A\({}_{10}C_4(e^6)(f)^4\)
B\({}_{10}C_3(e^7)(f)^3\)
C\({}_{10}C_3(e^7)(2f)^3\)
D\({}_{10}C_4(e^6)(2f)^4\)
The expression we derived matches option c exactly.
Examples
Binomial expansion is a fundamental tool in probability and statistics, particularly when calculating binomial distributions. For example, if a company wants to estimate the likelihood of exactly 3 successes out of 10 independent trials where each trial has a certain probability, the binomial theorem allows for efficient calculation of the probability distribution. Understanding how to extract individual terms helps in identifying specific outcomes in such experiments.