Combinations

Maths · Permutation, combination and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths topics

The idea

A combination is a selection of objects in which the order does not matter. Choosing Ali, Sara and Bilal for a committee is one combination, whatever order you name them in. The number of combinations of n different objects taken r at a time is written ⁿC_r.

To decide between a permutation and a combination, ask one question: does changing the order give a different result? Arranging, seating, forming numbers and awarding first, second and third places are permutations. Choosing a team, a committee, or points to join are combinations.

Each selection of r objects can be arranged in r! ways, which is why ⁿP_r is r! times ⁿC_r.

Rules to remember

ⁿC_r = n!/(r! × (n − r)!)
ⁿC₀ = ⁿC_n = 1 and ⁿC₁ = n
Complementary combinations: ⁿC_r = ⁿC_n−r, so if ⁿC_p = ⁿC_q with p ≠ q, then n = p + q
ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r
ⁿP_r = r! × ⁿC_r
Selections from two groups are multiplied, e.g. 2 from 5 and 3 from 6: ⁵C₂ × ⁶C₃
n points, no three collinear: lines = ⁿC₂, triangles = ⁿC₃; diagonals of an n-sided polygon = ⁿC₂ − n

Common mistake

Students use ⁿP_r when only a selection is asked for, which counts each group r! times. If the order of the chosen objects makes no difference, use ⁿC_r.

Worked example

A cricket team of 11 is to be chosen from 13 players. In how many ways can this be done if the captain must be in the team?

The captain is already in, so choose the other 10 players from the remaining 12.
Number of ways = ¹²C₁₀ = ¹²C₂.
¹²C₂ = (12 × 11)/(2 × 1) = 66.

Answer: 66 ways

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Keep going

← Factorials and permutations Probability →Take a Maths test