Combinations

Counting unordered selections with nCr, its formula and symmetry, and using it for committees, teams chosen from groups, and at least or at most questions.

Unordered selections

Choosing $3$ people from $5$ for a team: the ordered count is ${}^5P_3 = 60$, but each team of $3$ has been counted $3! = 6$ times, once for each order. So there are $\dfrac{60}{6} = 10$ teams.

Example 1●

Evaluate ${}^{10}C_3$ and ${}^{10}C_7$.

${}^{10}C_3 = \dfrac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

${}^{10}C_7 = {}^{10}C_3 = 120$ by symmetry.

Committees with conditions

Example 2●●

A committee of $4$ is chosen from $6$ women and $5$ men. How many committees have exactly $2$ women?

${}^6C_2 \times {}^5C_2 = 15 \times 10 = 150$

Example 3●●●

With the same people, how many committees have at least one man?

All committees: ${}^{11}C_4 = 330$.

No men (all women): ${}^6C_4 = 15$.

At least one man: $330 - 15 = 315$.

Example 4●●●

How many committees of $4$ include a particular woman, Wei, but not a particular man, Sam?

${}^9C_3 = 84$

Wei is in. Sam is out, leaving 9 people for the other 3 places.

Flashcards

${}^nC_r$

$\dfrac{n!}{r!(n - r)!}$

${}^nC_r$ vs ${}^nP_r$

${}^nC_r = \dfrac{{}^nP_r}{r!}$: divide out the orders.

Symmetry

${}^nC_r = {}^nC_{n-r}$

At least one

Total minus none.

Practice

How many teams of $3$ can be chosen from $5$ people?
Evaluate ${}^{10}C_7$.
A committee of $4$ from $6$ women and $5$ men. How many have exactly $2$ women?
How many of those committees have at least one man?
How many committees of $4$ include Wei but not Sam?
Which is equal to ${}^{12}C_5$?
Evaluate ${}^8C_0$.
How many $5$-card hands can be dealt from a $52$-card pack?
Choosing a captain and vice-captain from $10$ players is counted by:
Ten people each shake hands once with every other person. How many handshakes?