Factorials, the fundamental counting principle, nPr and nCr, arrangements, selections, and basic probability, with worked examples and practice
Counting problems and the chance built on them. The single decision that settles almost every question is: does order matter? If yes, it is a permutation (arrangement); if no, it is a combination (selection). Probability then divides favourable outcomes by total outcomes. This sits on the multiplication and factorial ideas in number system and simplification.
The factorial n! is the product of all whole numbers from 1 to n. By convention 0! = 1 and 1! = 1.
| n | n! |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
Fundamental counting principle: if one task can be done in m ways and a second in n ways, the two together can be done in m times n ways. Add when the choices are alternatives ("or"); multiply when they happen in sequence ("and").
| Concept | Formula |
|---|---|
| Permutations of n distinct items taken r at a time | nPr = n! / (n minus r)! |
| Combinations of n distinct items taken r at a time | nCr = n! / [ r! times (n minus r)! ] |
| Link between them | nPr = nCr times r! |
| Arrange all n distinct items | n! |
| Arrange n items with repeats | n! / (p! times q! ...) for repeated groups of size p, q, ... |
| Circular arrangement of n items | (n minus 1)! |
| nC0 and nCn | both equal 1 |
| nCr equals nC(n minus r) | symmetry property |
| Concept | Rule |
|---|---|
| Probability of an event E | P(E) = favourable outcomes / total outcomes |
| Range | 0 (impossible) to 1 (certain) |
| Complement | P(not E) = 1 minus P(E) |
| Mutually exclusive (cannot both happen) | P(A or B) = P(A) + P(B) |
| Not mutually exclusive | P(A or B) = P(A) + P(B) minus P(A and B) |
| Independent events | P(A and B) = P(A) times P(B) |
Standard sample spaces: a coin has 2 outcomes, a die has 6, a standard deck has 52 cards (26 red, 26 black, 4 suits of 13, 12 face cards counting J, Q, K).
In how many ways can 5 recruits stand in a row for a photo?
All 5 distinct items arranged: 5! = 120 ways.
From 8 officers, a committee of 3 is to be chosen. How many committees?
8C3 = 8! / (3! times 5!) = (8 times 7 times 6) / (3 times 2 times 1) = 336 / 6 = 56.
How many distinct arrangements of the letters of the word LEVEL?
LEVEL has 5 letters: L appears twice, E appears twice, V once. Arrangements = 5! / (2! times 2!) = 120 / 4 = 30.
In how many ways can 6 guests be seated around a round table?
Circular arrangement of 6 = (6 minus 1)! = 5! = 120.
A fair die is rolled. What is the probability of getting a prime number?
Primes on a die are 2, 3, 5, so 3 favourable outcomes out of 6. P = 3/6 = 1/2.
One card is drawn from a standard pack of 52. What is the probability it is a king or a heart?
Kings = 4, hearts = 13, but the king of hearts is counted in both, so subtract 1. P = 4/52 + 13/52 minus 1/52 = 16/52 = 4/13.
Two fair coins are tossed. What is the probability of two heads?
Each head has probability 1/2; independent, so P = 1/2 times 1/2 = 1/4.