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Answer :
- Use the binomial theorem to express the general term: $T_{k+1} = \binom{n}{k} a^{n-k} b^k$.
- Identify that we want to find $T_6$ where $n=9$ and $k=5$.
- Calculate the binomial coefficient: $\binom{9}{5} = 126$.
- The 6th term is therefore $\boxed{126 a^4 b^5}$.
### Explanation
1. Understanding the Binomial Theorem
We are asked to find the 6th term in the expansion of $(a+b)^9$. The binomial theorem tells us how to expand expressions of the form $(x+y)^n$. Each term in the expansion has the form $\binom{n}{k}x^{n-k}y^k$, where $\binom{n}{k}$ is a binomial coefficient, also known as 'n choose k'.
2. Identifying the Correct Term
The general term in the binomial expansion of $(a+b)^n$ is given by $T_{k+1} = \binom{n}{k} a^{n-k} b^k$, where $k = 0, 1, 2, ..., n$. In our case, $n=9$ and we want to find the 6th term, which corresponds to $k+1 = 6$, so $k=5$.
3. Applying the Formula
Now we can plug in the values $n=9$ and $k=5$ into the formula for the term $T_{k+1}$: $$T_6 = \binom{9}{5} a^{9-5} b^5 = \binom{9}{5} a^4 b^5$$
4. Calculating the Binomial Coefficient
We need to calculate the binomial coefficient $\binom{9}{5}$. This is calculated as:$$\binom{9}{5} = \frac{9!}{5! (9-5)!} = \frac{9!}{5! 4!} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5! \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$$
5. Finding the 6th Term
Therefore, the 6th term in the expansion of $(a+b)^9$ is $126 a^4 b^5$.
6. Final Answer
Comparing our result with the given options, we see that the correct answer is $126 a^4 b^5$.
### Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of specific outcomes in a series of independent trials (like coin flips). They also appear in calculus when approximating functions using Taylor series. Understanding binomial expansions helps in fields like statistics, physics, and engineering where predicting probabilities and approximating complex functions are essential.
- Identify that we want to find $T_6$ where $n=9$ and $k=5$.
- Calculate the binomial coefficient: $\binom{9}{5} = 126$.
- The 6th term is therefore $\boxed{126 a^4 b^5}$.
### Explanation
1. Understanding the Binomial Theorem
We are asked to find the 6th term in the expansion of $(a+b)^9$. The binomial theorem tells us how to expand expressions of the form $(x+y)^n$. Each term in the expansion has the form $\binom{n}{k}x^{n-k}y^k$, where $\binom{n}{k}$ is a binomial coefficient, also known as 'n choose k'.
2. Identifying the Correct Term
The general term in the binomial expansion of $(a+b)^n$ is given by $T_{k+1} = \binom{n}{k} a^{n-k} b^k$, where $k = 0, 1, 2, ..., n$. In our case, $n=9$ and we want to find the 6th term, which corresponds to $k+1 = 6$, so $k=5$.
3. Applying the Formula
Now we can plug in the values $n=9$ and $k=5$ into the formula for the term $T_{k+1}$: $$T_6 = \binom{9}{5} a^{9-5} b^5 = \binom{9}{5} a^4 b^5$$
4. Calculating the Binomial Coefficient
We need to calculate the binomial coefficient $\binom{9}{5}$. This is calculated as:$$\binom{9}{5} = \frac{9!}{5! (9-5)!} = \frac{9!}{5! 4!} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5! \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$$
5. Finding the 6th Term
Therefore, the 6th term in the expansion of $(a+b)^9$ is $126 a^4 b^5$.
6. Final Answer
Comparing our result with the given options, we see that the correct answer is $126 a^4 b^5$.
### Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of specific outcomes in a series of independent trials (like coin flips). They also appear in calculus when approximating functions using Taylor series. Understanding binomial expansions helps in fields like statistics, physics, and engineering where predicting probabilities and approximating complex functions are essential.
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