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Permutation with repetition

When order matters and an object can be chosen more than once then the number of

permutations is:

n r

where n is the number of objects from which you can choose and r is the number to be chosen.

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three letter patterns (trigrams) you find that there are 4 3 or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total.

Applications

The binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

( x + y ) n = k = 0 n n k x k y n - k

Whenever n is a positive integer, the numbers

n k = n ! k ! ( n - k ) !

are the binomial coefficients (the coefficients in front of the powers).

For example, here are the cases n = 2, n = 3 and n = 4:

( x + y ) 2 = x 2 + 2 x y + y 2 ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4

The coefficients form a triangle, where each number is the sum of the two numbers above it:

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayyám in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.

The number plate on a car consists of any 3 letters of the alphabet (excluding the vowels and 'Q'), followed by any 3 digits (0 to 9). For a car chosen at random, what is the probability that the number plate starts with a 'Y' and ends with an odd digit?

  1. The number plate starts with a 'Y', so there is only 1 choice for the first letter, and ends with an even digit, so there are 5 choices for the last digit (1, 3, 5, 7, 9).

  2. Use the counting principle. For each of the other letters, there are 20 possible choices (26 in the alphabet, minus 5 vowels and 'Q') and 10 possible choices for each of the other digits.

    Number of events = 1 × 20 × 20 × 10 × 10 × 5 = 200 000

  3. Use the counting principle. This time, the first letter and last digit can be anything.

    Total number of choices = 20 × 20 × 20 × 10 × 10 × 10 = 8 000 000

  4. The probability is the number of events we are counting, divided by the total number of choices.

    Probability = 200 000 8 000 000 = 1 40 = 0 , 025

Show that

n ! ( n - 1 ) ! = n
  1. Method 1: Expand the factorial notation.

    n ! ( n - 1 ) ! = n × ( n - 1 ) × ( n - 2 ) × . . . × 2 × 1 ( n - 1 ) × ( n - 2 ) × . . . × 2 × 1

    Cancelling the common factor of ( n - 1 ) × ( n - 2 ) × . . . × 2 × 1 on the top and bottom leaves n .

    So n ! ( n - 1 ) ! = n

  2. Method 2: We know that P ( n , r ) = n ! ( n - r ) ! is the number of permutations of r objects, taken from a pool of n objects. In this case, r = 1 . To choose 1 object from n objects, there are n choices.

    So n ! ( n - 1 ) ! = n

Exercises

  1. Tshepo and Sally go to a restaurant, where the menu is:
    Starter Main Course Dessert
    Chicken wings Beef burger Chocolate ice cream
    Mushroom soup Chicken burger Strawberry ice cream
    Greek salad Chicken curry Apple crumble
    Lamb curry Chocolate mousse
    Vegetable lasagne
    1. How many different combinations (of starter, main course, and dessert) can Tshepo have?
    2. Sally doesn't like chicken. How many different combinations can she have?
  2. Four coins are thrown, and the outcomes recorded. How many different ways are there of getting three heads? First write out the possibilities, and then use the formula for combinations.
  3. The answers in a multiple choice test can be A, B, C, D, or E. In a test of 12 questions, how many different ways are there of answering the test?
  4. A girl has 4 dresses, 2 necklaces, and 3 handbags.
    1. How many different choices of outfit (dress, necklace and handbag) does she have?
    2. She now buys two pairs of shoes. How many choices of outfit (dress, necklace, handbag and shoes) does she now have?
  5. In a soccer tournament of 9 teams, every team plays every other team.
    1. How many matches are there in the tournament?
    2. If there are 5 boys' teams and 4 girls' teams, what is the probability that the first match will be played between 2 girls' teams?
  6. The letters of the word 'BLUE' are rearranged randomly. How many new words (a word is any combination of letters) can be made?
  7. The letters of the word 'CHEMISTRY' are arranged randomly to form a new word. What is the probability that the word will start and end with a vowel?
  8. There are 2 History classes, 5 Accounting classes, and 4 Mathematics classes at school. Luke wants to do all three subjects. How many possible combinations of classes are there?
  9. A school netball team has 8 members. How many ways are there to choose a captain, vice-captain, and reserve?
  10. A class has 15 boys and 10 girls. A debating team of 4 boys and 6 girls must be chosen. How many ways can this be done?
  11. A secret pin number is 3 characters long, and can use any digit (0 to 9) or any letter of the alphabet. Repeated characters are allowed. How many possible combinations are there?

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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