Algebra

1. Introduction

Grade 12 algebra brings together counting, series, structure and the number system. It opens with the systematic counting of arrangements and selections (permutations and combinations), uses these counts to build the binomial theorem, and extends the theorem to any index so that important functions such as $e^x$, $a^x$ and $\log(1+x)$ can be expanded as series. It then studies the abstract idea of a group, which captures what addition and multiplication have in common, applies De Moivre's theorem to extract roots of complex numbers, and closes with polynomial and quadratic equations, mathematical induction, and the solution of linear systems by Cramer's rule and matrices. Throughout, the emphasis is on being able to compute confidently and to justify each result.

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2. Permutations and combinations

The fundamental principle of counting

If one task can be done in $m$ ways and, after it, a second independent task can be done in $n$ ways, then the two tasks together can be done in $m \times n$ ways. This multiplication principle extends to any number of successive choices, and it underlies every counting formula below.

Worked example. How many three digit numbers can be formed from the digits $1,2,3,4,5$ without repetition? The hundreds place can be filled in $5$ ways, the tens in the remaining $4$ ways, the units in $3$ ways, giving $5\times 4\times 3 = 60$ numbers.

Permutations

A permutation is an arrangement in which order matters. The number of arrangements of $n$ distinct objects taken $r$ at a time is

$$^{n}P_{r} = \frac{n!}{(n-r)!}, \qquad 0 \le r \le n.$$

In particular $^{n}P_{n} = n!$. When some objects are alike, we divide by the factorials of the repeated groups: the number of arrangements of $n$ objects of which $p$ are of one kind, $q$ of another and $r$ of a third is

$$\frac{n!}{p!,q!,r!}.$$

For a circular arrangement of $n$ distinct objects the number of ways is $(n-1)!$, because one object may be fixed to remove the rotational repetition; if clockwise and anticlockwise arrangements are treated as the same (as with a garland), the count is $\tfrac12 (n-1)!$.

Worked example. How many distinct arrangements are there of the letters of the word MATHEMATICS? There are $11$ letters with M, A and T each repeated twice, so the number of arrangements is

$$ \begin{aligned} \frac{11!}{2!,2!,2!} &= \frac{39916800}{8} \ &= 4989600. \end{aligned} $$

Combinations

A combination is a selection in which order does not matter. The number of selections of $r$ objects from $n$ distinct objects is

$$ \begin{aligned} ^{n}C_{r} &= \binom{n}{r} \ &= \frac{n!}{r!,(n-r)!}. \end{aligned} $$

Two properties are used constantly:

$$ \begin{aligned} \binom{n}{r} &= \binom{n}{n-r} \ \qquad \binom{n}{r} + \binom{n}{r-1} &= \binom{n+1}{r}. \end{aligned} $$

The second is Pascal's rule, which builds Pascal's triangle. Since order matters for permutations but not for combinations, the two are linked by $^{n}P_{r} = r!,\binom{n}{r}$.

Worked example. From $7$ men and $5$ women, in how many ways can a committee of $3$ men and $2$ women be formed? The men can be chosen in $\binom{7}{3} = 35$ ways and the women in $\binom{5}{2} = 10$ ways, so by the multiplication principle there are $35 \times 10 = 350$ committees.

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3. Binomial theorem for a positive integral index

For any positive integer $n$,

$$ \begin{aligned} (a+b)^n &= \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r} \ &= \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \cdots + \binom{n}{n}b^n. \end{aligned} $$

The coefficients $\binom{n}{r}$ are the binomial coefficients and coincide with the rows of Pascal's triangle.

Proof by induction. The statement is true for $n=1$ since $(a+b)^1 = a+b$. Assume it holds for $n=k$. Then

$$ \begin{aligned} (a+b)^{k+1} &= (a+b)(a+b)^k \ &= (a+b)\sum_{r=0}^{k}\binom{k}{r}a^{k-r}b^{r}. \end{aligned} $$

Multiplying out and collecting the term in $a^{k+1-r}b^{r}$ uses Pascal's rule $\binom{k}{r}+\binom{k}{r-1}=\binom{k+1}{r}$, which gives exactly

$$(a+b)^{k+1} = \sum_{r=0}^{k+1}\binom{k+1}{r}a^{k+1-r}b^{r}.$$

Hence the result holds for $k+1$, and by induction for all positive integers $n$.