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Formula Sheets / Maths

Chapter 6

Binomial Theorem

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Expansion and general term Binomial coefficients Any index

Expansion and general term

Binomial theorem
$(x + a)^{n} = r = 0 \sum n^{n} C_{r} x^{n - r} a^{r}$
General term
$T_{r + 1} =^{n} C_{r} x^{n - r} a^{r}$
Middle term
$n$ even: $T_{\frac{n}{2} + 1};$ $n$ odd: $T_{\frac{n + 1}{2}}, T_{\frac{n + 3}{2}}$
Term independent of x
put the power of $x$ in $T_{r + 1}$ equal to $0$
Greatest coefficient
$n$ even: $^{n} C_{n /2};$ $n$ odd: $^{n} C_{(n - 1) /2} =^{n} C_{(n + 1) /2}$

Binomial coefficients

Sums
$r = 0 \sum n^{n} C_{r} = 2^{n},$ $C_{0} + C_{2} + \dots = C_{1} + C_{3} + \dots = 2^{n - 1}$
Weighted sum
$r = 0 \sum n r^{n} C_{r} = n 2^{n - 1}$
Squares
$r = 0 \sum n (^{n} C_{r})^{2} =^{2 n} C_{n}$
Vandermonde
$r \sum^{m} C_{r}^{n} C_{k - r} =^{m + n} C_{k}$
Number of terms
$(x + a)^{n} : n + 1;$ $(x_{1} + \dots + x_{r})^{n} :^{n + r - 1} C_{r - 1}$

Any index

$(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots,$ $∣ x ∣ < 1, n \in Q$

Permutations and Combinations Matrices and Determinants