Choosing an Outfit

3 pairs of pants (B, R, O)

2 tops (W, R)

2 shoes (B, W)

- Tree Diagram

# of outfits = 3 × 2 × 2 = 12

2.How many 3-digit numbers can be formed if:

- there are no restrictions?
**ANS**: 9 × 10 × 10 = 900 (a 3 digit number can’t start with zero) - there are no repetitions?
**ANS**: 9 × 9 × 8 = 648 (gain the zero, lose the 1st digit number) - How many 4-letter arrangements can be formed if: (26 letters in the alphabet?)
- there are no repetitions?
**ANS**: 26 × 25 × 24 × 23 = 358 800 - repetitions allowed?
**ANS**: 264 = 456 976 a.k.a. 26 x 26 x 26 x 26 = - How many numbers can be formed from the digits 1, 4, 6, and 8 (one of each) if the number:
- Must be even
**ANS:**3 × 2 × 1 ×**3**(do first) = 18 - Must be greater than 5 000.
**ANS:**2 × 3 × 2 × 1 = 12 - Must be greater than 400.
**ANS:**3 × 3 × 2 + 4 × 3 × 2 × 1 = 18 + 24 = 42 - How many 5-letter arrangements can be made from the letters A, B, C, D, E, F, G?

**ANS**: 7P5 or 7 × 6 × 5 × 4 × 3 = 2 520

- How many ways can 6 books be arranged on a shelf?

**ANS**: 6P6 or 6 × 5 × 4 × 3 × 2 × 1 = 720

- 5-letter arrangements are formed from the letters in
**NUMBERS**. How many if: - no vowels?
**ANS**: 5 × 4 × 3 × 2 × 1 = 120 - 1st and last letters are consonants?
**ANS**: 5 × 5 × 4 × 3 × 4 = 1 200

___ x ___ x ___ x ___ x ____ AEIOU are vowels in the alphabet

Con Any Any Any Con

- 6-letter arrangements are formed from
**ABCDEF**. How many if:

- no restrictions?

**ANS**: 6 × 5 × 4 × 3 × 2 × 1 = 6! = 720

- A & B together? (think of AB as one letter)

**ANS**: 5 × 4 × 3 × 2 × 1 × 2 × 1 = 120

Number of ways to arrange AB ↵

The 2 × 1 is the placing of the A and B

The 5 × 4 × 3 × 2 × 1 are the placing of the AB as one unit and the CDEF as 4 units.

- A, B, & C together?
**ANS**: 4 × 3 × 2 × 1 × 3 × 2 × 1 = 144

The 4 × 3 × 2 × 1 is the arranging of DEF (in any order) and the ABC as a whole.

The 3 × 2 × 1 is the arranging of the ABC.

- A & B separated? (arrange others first)
**ANS**: 4 × 3 × 2 × 1 × 5 × 4 = 480 OR #a – #b

The 4 × 3 × 2 × 1 is the CDEF

The 5 is the A or B (one or the other) and the CDEF

You basically have __ C __ D __ E __ F ___

Now the CDEF can go in any order hence the 4!

The A or B could go in any of the 5 spaces hence the multiplying 5

Once the A or B is placed, the other one has 4 difference places to go, hence the multiplying 4.

- A, B, & C separated?
**ANS**: 3 × 2 × 1 × 4 × 3 × 2 = 144 (can’t do as above)

Same hint as above. You basically have __ D __ E __ F ___

Now the DEF can go in any order hence the 4!

The A or B or C could go in any of the 4 spaces hence the multiplying 4

Once the A B or C is placed, the other two have 3 difference places to go, hence the multiplying 3.

The multiplying 2 is the last A, B or C to go in either of the empty two spots.

- the word BAD appears?
**ANS**: 4 × 3 × 2 × 1 = 24

The 4 is the word BAD or C, E, or F (4 items as BAD is one unit)

The 3 is the 4 options above less the one already used.

The 2 is the 3 options above less the one already used.

The 1 is the only option left.

BAD cannot be switched as it must remain as BAD

- How many arrangements of the letters in the word:

- BANANA

**ANS**: 3 A’s and 2 N’s

- STATISTICS

**ANS**: 3 S’s, 3 T’s and 2 I’s

Computation

A committee of 3 is selected from 5 girls and 7 boys. How many if:

- no restrictions?
**ANS**: 12C3 = 220 - all boys?
**ANS**: 7C3 = 35 - 1 girl and 2 boys?
**ANS**: 5C1 ∙ 7C2 = 105 - At least 1 girl?
**ANS**: 1 girl or 2 girls or 3 girls = 5C1 ∙ 7C2 + 5C2 ∙ 7C1 + 5C3

= 105 + 70 + 10

= 185

OR #a − #b = 220 − 35

= 185

- 10 points are on a circle (no 3 are collinear). How many:

- Different lines are determined?

**ANS**: 10C2 = 45

A line is a connection of 2 points. So, take the 10 points and connect 2 of them.

- quadrilaterals?

**ANS**: 10C4 = 210

A quadrilateral has 4 sides. So, take 10 points and connect 4 of them.