UNIT 7 • ALGEBRA

Permutations & Combinations
nPr, nCr and Factorials

Permutations count arrangements, where order matters: ${}^nP_r = \frac{n!}{(n - r)!}$ . Combinations count selections, where it doesn't: ${}^nC_r = \frac{n!}{r!\,(n - r)!}$ .

THE TWO FORMULAS0 ≤ r ≤ n

\begin{gathered} {}^nP_r = \frac{n!}{(n - r)!} \qquad {}^nC_r = \frac{n!}{r!\,(n - r)!} \\ n! = n \times (n - 1) \times \cdots \times 2 \times 1 \end{gathered}

1. Does Order Matter?Ranks, roles, PINs, seating: yes (P). Teams, hands of cards, groups: no (C).

2. Repeats Allowed?Can the same item be used twice? Then it is

n^r

arrangements instead.

3. Cancel FirstCancel the factorials before multiplying:

\frac{10!}{7!} = 10 \times 9 \times 8

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A class of 10 students must pick 3 representatives. (a) In how many ways can a team of 3 be chosen? (b) In how many ways if the 3 are a president, a secretary and a treasurer?

STEP 01 • (a) ORDER DOESN'T MATTER: COMBINATIONS

{}^{10}C_3 = \frac{10!}{3!\,7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120

STEP 02 • (b) DIFFERENT ROLES: PERMUTATIONS

Now Asha as president and Ben as secretary differs from the other way round, so order matters:

{}^{10}P_3 = \frac{10!}{7!} = 10 \times 9 \times 8 = 720

STEP 03 • HOW THEY CONNECT

{}^{10}P_3 = 3! \times {}^{10}C_3 = 6 \times 120 = 720

Each team of 3 can hand out the 3 roles in 3! = 6 ways.

FINAL CONCLUSION

(a) 120 teams, (b) 720 ways with roles

{}^{10}C_3 = 120, \qquad {}^{10}P_3 = 720

The calculator below opens on 10C3 and works with exact answers however big.

LABORATORY

Interactive Permutations & Combinations Calculator

LABORATORY

Counting Engine

What do you want to count?

Total nChosen r

Common Questions

What is the difference between a permutation and a combination?

A permutation is an arrangement, where order matters (first, second, third place). A combination is a selection, where order doesn't matter (a team of three). Each combination of r things gives r! permutations, so

{}^nP_r = r! \times {}^nC_r

Why is 0! equal to 1?

There is exactly one way to arrange nothing. It also keeps the formulas working:

{}^nC_n = \frac{n!}{n!\,0!} = 1

How many ways can n people sit around a round table?

(n - 1)!

: fix one person and arrange the rest, because turning the whole table doesn't make a new seating. For beads on a necklace, which can also be flipped, it is

\frac{(n - 1)!}{2}

How do you count arrangements of a word with repeated letters?

Divide by the factorial of each repeat: MISSISSIPPI has 11 letters with I×4, S×4 and P×2, so

\frac{11!}{4!\,4!\,2!} = 34{,}650

Related Methods

AP & GP Quadratic Equation

PreviousAP & GP

Unit 7•50 of 63

NextDeterminant

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 7 • ALGEBRA

Permutations & Combinations
nPr, nCr and Factorials

Permutations count arrangements, where order matters: ${}^nP_r = \frac{n!}{(n - r)!}$ . Combinations count selections, where it doesn't: ${}^nC_r = \frac{n!}{r!\,(n - r)!}$ .

THE TWO FORMULAS0 ≤ r ≤ n

\begin{gathered} {}^nP_r = \frac{n!}{(n - r)!} \qquad {}^nC_r = \frac{n!}{r!\,(n - r)!} \\ n! = n \times (n - 1) \times \cdots \times 2 \times 1 \end{gathered}

1. Does Order Matter?Ranks, roles, PINs, seating: yes (P). Teams, hands of cards, groups: no (C).

2. Repeats Allowed?Can the same item be used twice? Then it is

n^r

arrangements instead.

3. Cancel FirstCancel the factorials before multiplying:

\frac{10!}{7!} = 10 \times 9 \times 8

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A class of 10 students must pick 3 representatives. (a) In how many ways can a team of 3 be chosen? (b) In how many ways if the 3 are a president, a secretary and a treasurer?

STEP 01 • (a) ORDER DOESN'T MATTER: COMBINATIONS

{}^{10}C_3 = \frac{10!}{3!\,7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120

STEP 02 • (b) DIFFERENT ROLES: PERMUTATIONS

Now Asha as president and Ben as secretary differs from the other way round, so order matters:

{}^{10}P_3 = \frac{10!}{7!} = 10 \times 9 \times 8 = 720

STEP 03 • HOW THEY CONNECT

{}^{10}P_3 = 3! \times {}^{10}C_3 = 6 \times 120 = 720

Each team of 3 can hand out the 3 roles in 3! = 6 ways.

FINAL CONCLUSION

(a) 120 teams, (b) 720 ways with roles

{}^{10}C_3 = 120, \qquad {}^{10}P_3 = 720

The calculator below opens on 10C3 and works with exact answers however big.

LABORATORY

Interactive Permutations & Combinations Calculator

LABORATORY

Counting Engine

What do you want to count?

Total nChosen r

Common Questions

What is the difference between a permutation and a combination?

{}^nP_r = r! \times {}^nC_r

Why is 0! equal to 1?

There is exactly one way to arrange nothing. It also keeps the formulas working:

{}^nC_n = \frac{n!}{n!\,0!} = 1

How many ways can n people sit around a round table?

(n - 1)!

: fix one person and arrange the rest, because turning the whole table doesn't make a new seating. For beads on a necklace, which can also be flipped, it is

\frac{(n - 1)!}{2}

How do you count arrangements of a word with repeated letters?

Divide by the factorial of each repeat: MISSISSIPPI has 11 letters with I×4, S×4 and P×2, so

\frac{11!}{4!\,4!\,2!} = 34{,}650

Related Methods

AP & GP Quadratic Equation

PreviousAP & GP

Unit 7•50 of 63

NextDeterminant

Permutations & Combinations nPr, nCr and Factorials

Step-by-Step Worked Example

(a) 120 teams, (b) 720 ways with roles

Interactive Permutations & Combinations Calculator

Counting Engine

Common Questions

NETZ Engine

Permutations & Combinations nPr, nCr and Factorials

Step-by-Step Worked Example

(a) 120 teams, (b) 720 ways with roles

Interactive Permutations & Combinations Calculator

Counting Engine

Common Questions

Permutations & Combinations
nPr, nCr and Factorials

Permutations & Combinations
nPr, nCr and Factorials