Binomial Theorem: Speed-Solving Techniques (6)

Question

Find the coefficient of $x^7$ in the expansion of $(2 + 3x)^{10}$.

Solution — Step by Step

Final answer: Coefficient of $x^7 = 2{,}099{,}520$.

Why This Works

The binomial theorem hands you the structure: every term in $(a+b)^n$ has the form $\binom{n}{r} a^{n-r} b^r$. The exponent of $b$ is $r$. Match the power of the variable to $r$, then plug in.

For $(2 + 3x)^{10}$, the $x^r$ coefficient picks up factors $2^{10-r}$, $3^r$, and $\binom{10}{r}$. Speed comes from skipping intermediate steps once you’ve memorised the pattern.

Alternative Method (Speed)

Mental shortcut: “coefficient of $x^7$ = $\binom{10}{7} \cdot 2^3 \cdot 3^7$”. Directly write this without listing terms. Then compute $\binom{10}{7} = 120$ and the powers. The answer falls out in 30 seconds.

Common Mistake

Mixing up which variable is which. $(2 + 3x)^{10}$ has $a = 2$ (constant), $b = 3x$. Some students write $a = 2x$ by accident and get $\binom{10}{7} (2x)^3 (3)^7$, which is wrong.