Chapter 9 Review

Chapter 9 Review

You'll need paper, pen/pencil, and a calculator

Combinatorics: The Art of Counting

• Combinations vs. Permutations: _nP_r, _nC_r

• Multiplication principle of counting: n · m

• When you can take it or leave it: the 2ⁿ approach

Multiple Choice

How would you solve?

From the 20 books you purchased this past year, you plan to take five with you on vacation. How many different sets of five books can you take?

• Permutation

• Combination

• 2ⁿ

• Multiplication Principle of Counting

Multiple Choice

An election ballot asks voters to select three city commissioners from a group of six candidates.  In how many ways can this be done?

• 6!
• 120
• 20
• 3!

Since the order doesn't matter when a committee is being chosen, we use combinations to solve this problem

₆C₃ = 6!/(6-3)!3! = 20

Multiple Choice

How many 2-digit numbers can you make using the digits 1, 2, 3, & 4 without repeating the digits?

• 90
• 100
• 12
• 24

4 · 3 = 12

Think of how many ways you can assign each digit

Multiple Choice

When buying a new car, you can choose up to 6 additional accessories. How many different ways can the customer order the car?

• ₆P₂

• 2⁶

• 6²

• ₆C₂

• 6·2

2 · 2 · 2 · 2 · 2 · 2 = 2⁶

Each accessory can be chosen two ways (take it or leave it); using MCP, you multiply 2 by itself 6 times

The Binomial Theorem

• Pascal's Triangle can also be used to expand binomials

Multiple Choice

Choose the right Pascal's triangle

Multiple Choice

Find the coefficient of the x³ in (2x+1)¹²

• 112640
• 440
• 1760
• 440x³

Multiple Choice

Find the 14th term of (2x-y)¹⁸

• -274,176x⁵y¹³

• 274,176x⁵y¹³

• 64,064x⁵y¹⁴

• -64,064x⁵y¹⁴

Sequences

• Explicitly vs. recursively defined sequences

• Arithmetic sequences: a_n = a₁ + d(n-1)

• Geometric sequences: a_n = a₁ (r^n-1)

• Converging sequences

Multiple Choice

Given the sequence:

125, 25, 5, ....

Write the rule a_n using the geometric sequence formula:

• a_n = (125)(1/5)^(n-1)

• a_n = 625(1/5)^(n-1)

• a_n = 625(5)^(n-1)

• a_n = 125 + 5n

Multiple Choice

Find the 22nd term of the following sequence:
5, 8, 11, ...

• 14
• 68
• 63
• 71

Series

• A series is a sum of terms in a sequence

• We often use Sigma notation to represent a series

• When we can find the sum of a series, the series converges

Multiple Choice

Write 1 + 4 + 9 + ... + 49

using sigma notation

• $\sum_{k=1}^7k^2$

• $\sum_{k=1}^8k^2$

• $\sum_{k=1}^{49}k^2$

Multiple Choice

Write 1 + 2 + 3 + ... + 10

using sigma notation

• $\sum_{d=1}^9d$

• $\sum_{d=1}^{10}d+1$

• $\sum_{d=1}^{10}d$

Multiple Choice

You have $10 in you bank account.  It doubles every month.  How much money will you have after 5 months?

• $50
• $35
• $320
• $250