13.5 Counting principles  (Page 3/12)

 Page 3 / 12

A family of five is having portraits taken. Use the Multiplication Principle to find the following.

How many ways can the family line up for the portrait?

120

How many ways can the photographer line up 3 family members?

60

How many ways can the family line up for the portrait if the parents are required to stand on each end?

12

Finding the number of permutations of n Distinct objects using a formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ objects and we want to choose $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ objects from the set in order, we write $\text{\hspace{0.17em}}P\left(n,r\right).\text{\hspace{0.17em}}$ Another way to write this is ${n}_{}{P}_{r},\text{\hspace{0.17em}}$ a notation commonly seen on computers and calculators. To calculate $\text{\hspace{0.17em}}P\left(n,r\right),\text{\hspace{0.17em}}$ we begin by finding $\text{\hspace{0.17em}}n!,\text{\hspace{0.17em}}$ the number of ways to line up all $n$ objects. We then divide by $\text{\hspace{0.17em}}\left(n-r\right)!\text{\hspace{0.17em}}$ to cancel out the $\text{\hspace{0.17em}}\left(n-r\right)\text{\hspace{0.17em}}$ items that we do not wish to line up.

Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is $6×5×4=120.$ Using factorials, we get the same result.

$\text{\hspace{0.17em}}\frac{6!}{3!}=\frac{6·5·4·3!}{3!}=6·5·4=120\text{\hspace{0.17em}}$

There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.

$\text{\hspace{0.17em}}P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}\text{\hspace{0.17em}}$

Note that the formula stills works if we are choosing all $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ objects and placing them in order. In that case we would be dividing by $\text{\hspace{0.17em}}\left(n-n\right)!\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}0!,\text{\hspace{0.17em}}$ which we said earlier is equal to 1. So the number of permutations of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ objects taken $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ at a time is $\text{\hspace{0.17em}}\frac{n!}{1}\text{\hspace{0.17em}}$ or just $\text{\hspace{0.17em}}n!\text{.}$

Formula for permutations of n Distinct objects

Given $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ distinct objects, the number of ways to select $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ objects from the set in order is

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Given a word problem, evaluate the possible permutations.

1. Identify $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ from the given information.
2. Identify $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ from the given information.
3. Replace $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ in the formula with the given values.
4. Evaluate.

Finding the number of permutations using the formula

A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?

Substitute $\text{\hspace{0.17em}}n=12\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=9\text{\hspace{0.17em}}$ into the permutation formula and simplify.

There are 79,833,600 possible permutations of exam questions!

Could we have solved [link] using the Multiplication Principle?

Yes. We could have multiplied $\text{\hspace{0.17em}}15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\text{\hspace{0.17em}}$ to find the same answer .

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

How many ways can the 7 actors line up?

$\text{\hspace{0.17em}}P\left(7,7\right)=5,040\text{\hspace{0.17em}}$

How many ways can 5 of the 7 actors be chosen to line up?

$\text{\hspace{0.17em}}P\left(7,5\right)=2,520\text{\hspace{0.17em}}$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions