The Binomial Theorem
What you must understand
You must state and apply the binomial theorem for positive integer powers, use the general term (\binom{n}{r}a^{n-r}b^r) to extract a particular term or coefficient, and extend to negative and fractional indices using the expansion of ((1+x)^n) valid for (|x|<1). You should also handle identities obtained by substituting values into expansions, and use binomial coefficients in counting and probability contexts.
Traps
- General term indexing: the term in (x^r) corresponds to (\binom{n}{r}) only when the expansion is written with matching powers - always derive the general term explicitly rather than quoting a formula from memory.
- Validity range: the expansion of ((1+x)^n) for non-integer (n) converges only for (|x|<1); the answer is incomplete without this condition.
- Coefficient versus term: questions asking for the coefficient exclude the power of (x); asking for the term includes it.
- Factorial manipulation: (\binom{n}{r} = \frac{n!}{r!(n-r)!}) with (0! = 1); errors with (n) negative or fractional require the generalised form.
- Substitution errors: when finding sums like (\sum \binom{n}{r}), substitute (x=1) correctly, and (x=-1) for alternating sums.
Rule of thumb
Write the general term first, simplify its power of (x), then set that power equal to the required value and solve for (r).
Worked example
Find the coefficient of (x^4) in ((2 + 3x)^7). The general term is (\binom{7}{r}2^{7-r}(3x)^r), so the power of (x) is (r). Setting (r=4): coefficient (= \binom{7}{4}2^{3}3^{4} = 35 \cdot 8 \cdot 81 = 22680).