Binomial expansion

The numbers $\left(\genfrac{}{}{0px}{}{n}{r}\right)$ (said “n choose r”, calculator key nCr) are the rows of Pascal’s triangle:

For small $n$, Pascal’s triangle by hand is fine; for $(2-\frac{x}{2}{)}^{10}$-style questions, nCr is faster and less error-prone. Symmetry helps too: $\left(\genfrac{}{}{0px}{}{n}{r}\right)=\left(\genfrac{}{}{0px}{}{n}{n-r}\right)$.

The term containing $b^r$ in $(a+b{)}^n$ is

Every “find the coefficient of $x^k$” question is this formula plus one line of index work: set the power of $x$ inside $b^r$ equal to $k$, solve for $r$, substitute. The critical discipline: $a$ and $b$ are the whole terms, coefficients and signs included. In $(2-3x{)}^5$, $a=2$ and $b=-3x$ — so $b^r$ carries $(-3{)}^r$, not just $x^r$.

The harder P1 variant runs the machine backwards: “the coefficient of $x^2$ in $(1+ax{)}^6$ is 60; find $a$.” Same general term:

Keep both roots unless the question restricts $a$ (“where $a>0$”). Examiner reports specifically call out dropped negative roots here. A second common variant multiplies two expansions — e.g. $(1+2x)(1-x{)}^7$ — where the target power collects from two products: constant × $x^k$ term plus $x$ term × $x^{k-1}$ term.

1. 1.Forgetting to raise b's coefficient to the power

   In $(2-3x{)}^5$, the $x^3$ term carries $(-3{)}^3=-27$, never $-3$. Bracket the whole term — $(-3x{)}^r$ — before separating number from letter.

2. 2.Losing the sign in (a − b)ⁿ

   Treat the minus as part of $b$: $b=-3x$. Odd powers of $b$ are negative, even powers positive — if your expansion doesn’t alternate, something’s wrong.

3. 3.Expanding the whole bracket when one term was asked

   “Find the coefficient of $x^3$” needs the general term and one value of $r$ — three lines. A full ten-term expansion invites arithmetic slips and burns minutes the paper doesn’t give back.

4. 4.Confusing 'coefficient' with 'term'

   The term is $-1080x^3$; the coefficient is $-1080$. Answer the noun the question used.

5. 5.Dropping the ± when solving for an unknown

   $a^2=4$ has two solutions. Write $a=\pm 2$, then discard one only if the question gives a reason to.