Inequalities, Permutation, and Combination: JAMB Mathematics Notes

Solving inequalities, and counting arrangements (permutation) or selections (combination) without listing every possibility.

Linear inequalities

Solve a linear inequality exactly like an equation, but flip the inequality sign whenever you multiply or divide both sides by a negative number.

Example 1: Solve $3 - 2 x \leq 9$ .

Subtract 3: $- 2 x \leq 6$ . Divide by $- 2$ (flip the sign): $x \geq - 3$ .

Quadratic inequalities

Rearrange to zero, factorize, find the critical values, then test which region satisfies the inequality.

Example 2: Solve $x^{2} - x - 6 > 0$ .

Factorize: $(x - 3) (x + 2) > 0$ . The critical values are $x = 3$ and $x = - 2$ . Testing regions shows the product is positive when $x < - 2$ or $x > 3$ .

Permutation

The number of ways to arrange $r$ items chosen from $n$ , where order matters: $n P r = \frac{n !}{( n - r )!}$ .

Example 3: In how many ways can 3 people be arranged in a queue of 5?

$5 P 3 = \frac{5 !}{( 5 - 3 )!} = \frac{120}{2} = 60$ .

Combination

The number of ways to choose $r$ items from $n$ , where order does not matter: $n C r = \frac{n !}{r ! ( n - r )!}$ .

Example 4: In how many ways can a committee of 3 be chosen from 8 people?

$8 C 3 = \frac{8 !}{3 ! \times 5 !} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$ .

Tip: if the question involves arranging (order matters -- a queue, a password, a race finish), use permutation. If it involves selecting a group (order doesn't matter -- a committee, a team), use combination.

Inequalities, Permutation, and Combination

Linear inequalities

Quadratic inequalities

Permutation

Combination

Key terms

Practise Inequalities, Permutation, and Combination questions

Related topics

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