# Combinations

### Combinations

How many choices do I have? This lesson can help you with answering this question! Combination is the process of selecting members from a set of items. Combination formula is your best friend in this lesson. Unlike permutations, order doesn't matter in combinations!

#### Lessons

? combination: nCr = number of selections of r items taken from a set of n distinct items (order does NOT matter!!)
= $\frac{{n!}}{{(n - r)!\;\;r!}}$
• 1.
Compare the difference between the following 2 scenarios:
a)
How many ways can a president, a vice president, and a treasurer be selected from a class of 20 students?

b)
How many different committees of 3 people can be selected from a class of 20 students?

• 2.
A standard deck of 52 cards consists of:
• 4 suits: diamonds, hearts, spades, and clubs
• each suit has 13 cards
• red cards: diamonds and hearts
• black cards: spades and clubs
• face cards: Jacks, Queens, and Kings
How many different 5-card hands can be formed containing:
a)
any 5 cards (no restrictions)?

b)
all black cards?

c)
1 black card and 4 red cards?

d)
all face cards?

e)
3 Jacks and 2 Kings?

f)
3 Jacks?

g)
flush (5 cards of the same suit)?

• 3.
? problems with "at least", "at most"
From a standard deck of 52 cards, how many different 5-card hands can be formed containing:
a)
exactly 2 Diamonds?

b)
at least 2 Diamonds?

c)
at most 2 Diamonds?

• 4.
There are 20 members in a student council, 7 boys and 13 girls. How many ways can a committee of 5 people be selected if there must be:
a)
at least 3 boys?

b)
at most 1 girl?

c)
at least 1 girl?

d)
at least 2 boys and 2 girls?

• 5.
Five points are marked on a circle. By connecting the points on the circle,
a)
How many triangles can be formed?

b)
How many line segments can be formed

c)
Form a pentagon inside the circle by connecting the 5 points. How many diagonals does the pentagon have?