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For the following exercises, assume that there are $n$ ways an event $A$ can happen, $m$ ways an event $B$ can happen, and that $A\text{and}B$ are non-overlapping.
Use the Addition Principle of counting to explain how many ways event $A\text{or}B$ can occur.
There are $\text{\hspace{0.17em}}m+n\text{\hspace{0.17em}}$ ways for either event $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ or event $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ to occur.
Use the Multiplication Principle of counting to explain how many ways event $\text{\hspace{0.17em}}A\text{and}B\text{\hspace{0.17em}}$ can occur.
Answer the following questions.
When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?
The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
Describe how the permutation of $n$ objects differs from the permutation of choosing $r$ objects from a set of $n$ objects. Include how each is calculated.
What is the term for the arrangement that selects $r$ objects from a set of $n$ objects when the order of the $r$ objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?
A combination; $\text{\hspace{0.17em}}C(n,r)=\frac{n!}{(n-r)!r!}\text{\hspace{0.17em}}$
For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
Let the set $A=\{-5,-3,-1,2,3,4,5,6\}.$ How many ways are there to choose a negative or an even number from $\mathrm{A?}$
Let the set $B=\{-23,-16,-7,-2,20,36,48,72\}.$ How many ways are there to choose a positive or an odd number from $A?$
$\text{\hspace{0.17em}}4+2=6\text{\hspace{0.17em}}$
How many ways are there to pick a red ace or a club from a standard card playing deck?
How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?
$\text{\hspace{0.17em}}5+4+7=16\text{\hspace{0.17em}}$
How many outcomes are possible from tossing a pair of coins?
How many outcomes are possible from tossing a coin and rolling a 6-sided die?
$\text{\hspace{0.17em}}2\times 6=12\text{\hspace{0.17em}}$
How many two-letter strings—the first letter from $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and the second letter from $\text{\hspace{0.17em}}B\u2014$ can be formed from the sets $\text{\hspace{0.17em}}A=\{b,c,d\}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B=\{a,e,i,o,u\}?\text{\hspace{0.17em}}$
How many ways are there to construct a string of 3 digits if numbers can be repeated?
$\text{\hspace{0.17em}}{10}^{3}=1000\text{\hspace{0.17em}}$
How many ways are there to construct a string of 3 digits if numbers cannot be repeated?
For the following exercises, compute the value of the expression.
$\text{\hspace{0.17em}}P(5,2)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(5,2)=20\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(8,4)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(3,3)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(3,3)=6\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(9,6)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(11,5)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}P(11,5)=55,440\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(8,5)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(12,4)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(12,4)=495\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(26,3)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(7,6)\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(7,6)=7\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}C(10,3)\text{\hspace{0.17em}}$
For the following exercises, find the number of subsets in each given set.
$\text{\hspace{0.17em}}\{1,2,3,4,5,6,7,8,9,10\}\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}{2}^{10}=1024\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}\{a,b,c,\dots ,z\}\text{\hspace{0.17em}}$
A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols
$\text{\hspace{0.17em}}{2}^{12}=4096\text{\hspace{0.17em}}$
The set of even numbers from 2 to 28
The set of two-digit numbers between 1 and 100 containing the digit 0
$\text{\hspace{0.17em}}{2}^{9}=512\text{\hspace{0.17em}}$
For the following exercises, find the distinct number of arrangements.
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