What is the fifth term in the expansion of $(3 x-3 y)^7$ ?

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05-11-2026
Mathematics

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What is the fifth term in the expansion of $(3 x-3 y)^7$ ?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T14:24:13-05:00

Identify $n=7$, $a=3x$, $b=-3y$, and the fifth term corresponds to $k=4$.
Apply the formula $T_5 = \binom{7}{4} (3x)^{3} (-3y)^4$.
Calculate the values: $\binom{7}{4}=35$, $(3x)^3=27x^3$, and $(-3y)^4=81y^4$.
Multiply the results to find the final coefficient: $35 \times 27 \times 81 = 76545$.
The final term is $\boxed{76545x^3 y^4}$.

Explanation

Identifying the components and the general term formula

To find the fifth term in the binomial expansion of $(3x - 3y)^7$, we apply the Binomial Theorem. The theorem states that $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$, where $\binom{n}{k}$ is the binomial coefficient given by $\frac{n!}{k!(n-k)!}$. In our expression, we have $a = 3x$, $b = -3y$, and $n = 7$. The terms are ordered such that the first term corresponds to $k=0$, the second to $k=1$, and so on. Therefore, the fifth term corresponds to $k = 4$. The general formula for the $(k+1)$-th term is $T_{k+1} = \binom{n}{k} a^{n-k} b^k$. For $k=4$, this is $T_5 = \binom{7}{4} (3x)^{7-4} (-3y)^4$.

Calculating the fifth term

First, we calculate the binomial coefficient $\binom{7}{4}$:
\[\binom{7}{4} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.\]
Next, we determine the powers of the individual terms:
\[(3x)^{7-4} = (3x)^3 = 3^3 x^3 = 27x^3.\]
\[(-3y)^4 = (-3)^4 y^4 = 81y^4.\]
Multiplying these components together, we get:
\[T_5 = 35 \times 27x^3 \times 81y^4.\]
Calculating the numerical coefficient:
\[35 \times 27 \times 81 = 35 \times 2187 = 76545.\]
Thus, the fifth term is $76545x^3 y^4$.

Examples

Binomial expansion is widely used in probability and statistics, particularly in calculating binomial distributions which model the number of successes in a series of independent experiments. For instance, if you are calculating the likelihood of specific outcomes in a series of coin flips or quality control tests, the binomial theorem allows you to determine the coefficient of a specific event combination, like getting exactly 4 heads out of 7 tosses. This mathematical structure helps engineers and scientists predict and analyze systems involving discrete events and binomial probabilities.