Probability · Ordinary Level

Permutations — Conditions

Ordinary Level · Class 12 · Tap NEXT to begin

Section 1 of 3

Fixed Positions

Fill the fussy box first

If a position has a rule (must start with a certain letter, must be odd…), fill that box first, then fill the rest with what’s left.

Worked example — the word DUBLIN

Using the letters of $DUBLIN$ (all different), how many $3$-letter words (i) in total; (ii) start with $D$; (iii) start with $D$ and end with a vowel?

(i) $6 \times 5 \times 4 = 120$

(ii) $1 \times 5 \times 4 = 20$

(iii) start $D$ ($1$), end a vowel $U$ or $I$ ($2$), middle from the other $4$: $1 \times 4 \times 2 = 8$

$120;\ 20;\ 8$

Worked example — five letters a–e

Using $a, b, c, d, e$ (each once), how many $5$-letter words (i) start with $a$; (ii) end with $e$; (iii) start with $a$ and end with $e$?

(i) $1 \times 4 \times 3 \times 2 \times 1 = 24$

(ii) $4 \times 3 \times 2 \times 1 \times 1 = 24$

(iii) $1 \times 3 \times 2 \times 1 \times 1 = 6$

$24;\ 24;\ 6$

Section 2 of 3

Number Conditions

Worked example — four-digit numbers

Using $3, 4, 5, 6$ (each once), how many $4$-digit numbers (i) in total; (ii) start with $4$; (iii) are over $5000$; (iv) are odd; (v) start with $3$ and are odd?

(i) $4! = 24$

(ii) $1 \times 3 \times 2 \times 1 = 6$

(iii) first digit $5$ or $6$: $2 \times 3 \times 2 \times 1 = 12$

(iv) last digit odd ($3$ or $5$): $3 \times 2 \times 1 \times 2 = 12$

(v) start $3$, then last odd is only $5$: $1 \times 2 \times 1 \times 1 = 2$

$24;\ 6;\ 12;\ 12;\ 2$

Section 3 of 3

Keeping Items Together

Bubble them together

To keep two items side by side, bubble them as one. Arrange the bubbles, then multiply by the ways inside the bubble ($PQ$ or $QP$ = $2$).

Worked example — p and q together

Using $p, q, r, s, t$ (each once), how many $5$-letter words have $p$ and $q$ always together?

Bubble $[pq]$ with $r, s, t$ = $4$ items to arrange: $4! = 24$

Inside the bubble $pq$ or $qp$ = $2$ ways

$2 \times 24 = 48$

$48$

Worked example — a and b together

Using $a, b, c$ (each once), how many $3$-letter words have $a$ and $b$ together? (Sample space check.)

Bubble $[ab]$ with $c$ = $2$ items: $2! = 2$; inside $= 2$

$2 \times 2 = 4$ — check: $abc, bac, cab, cba$

$4$

You try

Using $a, b, c, d, e$ (each once), find how many $5$-letter words have (i) $a$ and $b$ together; (ii) $a, b$ and $c$ together.

Bubble the letters, arrange the bubbles, then multiply by the ways inside the bubble.

(i) bubble $[ab]$ + $c, d, e$ = $4$ items: $4! \times 2 = 48$

(ii) bubble $[abc]$ + $d, e$ = $3$ items: $3! \times 3! = 6 \times 6 = 36$

$48;\ 36$

That’s Class 12 — the last one.

Conditions on positions, and keeping items together with the bubble method. You’ve finished Probability & Permutations!