Matrix Math
[[1,2],[6,-8]]+[[5,3],[-10,9]]
[[6,5],[-4,1]]
You own eight different books written by Charles Dickens. How many ways can a shelf arrangement of five of the Dickens books be formed?
P(8,5)=\frac{8!}{(8-5)!}=6720
In a city election there are four candidates for mayor, three candidates for vice-mayor, six candidates for treasurer, and two for secretary. In how many ways can there four offices be filled?
144 different ways
A Senate investigation subcommittee of four members is to be selected from a Senate committee of ten members. Determine the number of ways this can be done.
C(10,4)=\frac{10!}{4!(10-4)!}=210
Multiple payments made regularly at the end of the term
Ordinary Annuity
[[2,0],[-6,3]]-[[1,-4],[-4,3]]
[[1,4],[-2,0]]
From a pool of 10 job applicants, a list ranking the top 4 must be made. How many such lists are possible?
P(10,4)=\frac{10!}{(10-4)!}=5040
How many possible batting orders can the manager of a baseball team construct from nine available players?
362,880 possible batting orders
A bit is a 0 or a 1. A six-bit string is a sequence of length six consisting of 0s and 1s. How many six-bit strings contain exactly two 1s?
C(6,2)=15
Singular Matrix
A square matrix that does not have an inverse
[[3,5],[6,7]]+[[2,1],[9,0]]-[[3,4],[5,1]]
[[2,2],[10,6]]
How many distinguishable permutations can the letters ANTARCTICA form?
\frac{10!}{3!\cdot2!\cdot2!}=151,200
On a literature test there are 10 multiple-choice questions with 4 possible answers and 15 true-false questions. In how many possible ways can the 25 questions be answered?
34,359,738,368 different sets of test answers
How many different hands are possible in a bridge game? A bridge had consists of 13 cards dealt from a deck of 52 cards.
C(52,13)=635,013,559,600
A measure of how well the original data fits a straight line
Correlation Coefficient
[[-1,-3],[5,7]]\cdot[[3,2],[9,-10]]
[[-30,28],[78,-60]]
Weaver and Kline, a stock brokerage firm, has received nine inquiries regarding new accounts. In how many ways can these inquires be directed to three of the firm's account executives if each account executive is to handle three inquires?
\frac{9!}{3! \cdot 3! \cdot 3!}=1680
License plates in the state of Maryland consist of three letters of the alphabet followed by three digits. How many license plates will have all their digits distinct?
12,654,720 license plates with distinct digits. Note: there was no requirement that the letters be distinct.
The members of a string quartet comprising two violinists, a violist, and a cellist are to be selected from a group of six violinists, three violists, and two cellists, receptively. In how many ways can the string quartet be formed?
C(6,2)\cdot C(3,1) \cdot C(2,1) = 15 \cdot 3 \cdot 2=90
Note: We also needed to use the Multiplication Principle here.
An arrangement in which the order does matter
Permutation
Find the inverse for
[[3,-5],[2,-8]]
[[\frac{4}{7},\frac{-5}{14}],[\frac{1}{7},\frac{-3}{14}]]
You own eight books by Charles Dickens and six books by Mark Twain. You wish to display seven of the books on a shelf. If the first four positions are to be occupied by Dickens books and the last three by Twain books, in how many ways can this be done?
P(8,4)\cdotP(6,3)=201,600 \text{ ways}
The Multiplication Principle was also needed to solve this one!
Find the number of 7-digit telephone numbers with no repeated digits and the first number cannot be zero.
544,320 possible telephone numbers
From six women and four men a committee of three is to be formed. The committee must include at least two women. In how many ways can this be done?
C(6,2) \cdot C(4,1) + C(6,3) \cdot C(4,0) = 60+20=80
A \subseteq B
A is a subset of B. Every element in Set A can also be found in Set B.