Ximpledu

Number of Ways (Math)

See how to find the number of ways
(rule of sum, rule of product, complementary event).
6 examples and their solutions.

Rule of Sum

Formula

N = m + n + ...
To find the number of wanted ways (N),
add the possible number of ways (m, n, ...).

Example

Numbers from 1 to 10 are given.
Find the number of ways to pick an even number.
Solution

Example

A big die and a small die are tossed once.
Find the number of ways to get a sum of 9.
(The plural of die is dice.)

⚀ ⚁ ⚂ ⚃ ⚄ ⚅
Solution

Example

Find the number of shortest paths to move from A to B.

Solution

Rule of Product

Formula

N(A and B) = N(A)⋅N(B)⋅...
N(A and B): Number of ways of [A and B] happening
N(A): Number of ways of A happening
N(B): Number of ways of B happening
A and B don't affect each other.

Example

A fair die and a coin is tossed once.
Find the number of results.
Solution

Example

3 spoons, 5 cups, 2 dishes
Find the number of ways to pick a spoon, a cup, and a dish.
Solution

Complementary Event

Formula

N(A) = N(total) - N(not A)
N(not A): Number of ways of A not happening
(= complementary event)
In most cases, if there's [at least],
it's good to use this formula.

Example

A fair die is tossed twice.
Find the number of ways to get at least one multiple of 3.
Solution