Question

How many different three-ingredient sandwiches can be made if 8 ingredients are available?

Answer 1

8*7*5

Answer 2

Thank you (:

Answer 3

did you understand what did i do??

Answer 4

Yeah, pretty much. You did the permutations. But does the 6 reduce down to a 2? and then cancel? that's the only thing that got me.

Answer 5

I meant reduce down to a 3 lol now I think I'm just confusing myself.

Answer 6

experiment I'd love to know what you just did.

Answer 7

How many different three-ingredient sandwiches can be made if 8 ingredients are available?

Answer 8

Now, we are using nCr because order does not matter.

Answer 9

cheese, tomato, lettuce, is the same as tomato, cheese, lettuce, right?

Answer 10

nCr=(nPr)/r!
nPr=(n!)/(n-r)!
nCr=(n!)/[r!(n-r)!]

Answer 11

n=8
r=3

Answer 12

8!/[3!(8-3)!]
=8!/(3!5!)
=(8*7*6)/3!
=8*7

Answer 13

I honestly just don't understand how to use combinations or permutations. I'm a bit better at permuatations than combinations, but both confuse me. But I'm trying to follow through with what you are saying, so thank you lol.

Answer 14

Alright, let me explain just exactly what combinations and permutations are

Answer 15

Well, you understand permutations, right?

Answer 16

yeah, I'm pretty sure I do.

Answer 17

ok. Now, there is only a small difference between combinations and permutations.

Answer 18

in permutations, "order" matters.

Answer 19

Say, the winning number of a lottery ticket.

Answer 20

right

Answer 21

1119 is different from 9111

Answer 22

in combinations, however "order" does not matter.

Answer 23

if we were to take 2 random people from a group of 10 people, selecting Bob then Joe is the same as selecting Joe then Bob

Answer 24

Now, there are formulas for permuations and combinations, and unfortunately, these require memorization.

Answer 25

(they don't if you understand them intuitively, however)

Answer 26

alright, so I just have to memorize the permutations and combinations formula and try and learn how to do them?

Answer 27

Well, you can try to understand why they are what they are, and then try to memorize; that helps.

Answer 28

I'll try to explain...

Answer 29

Yeah I can do that. That's a good idea, thank you. But you don't have to explain it to me if you don't want to. haha but I'd appreciate if you did. :)

Answer 30

Ok, let's say we have a 8 digit password (0-9), and the numbers cannot be repeated.

Answer 31

In other words, that would be 10 choices for the first, 9 for the second (because one is taken up by the first), 8 for the third, 7 for the fourth, 6 for the fifth, 5 for the sixth, 4 for the seventh, and 3 for the eighth.

Answer 32

Now we multiply those all out.

Answer 33

10*9*8*7*6*5*4*3
I mean, multiply those by each other.

Answer 34

yes, so that would be 1,814,400 right?

Answer 35

Not sure. But, what's important is to notice it's striking resemblance to the factorial function. The two must be related.

Answer 36

in fact, we didn't select the last two digits for the password, right?

Answer 37

right

Answer 38

those last two digits happen to be 2!

Answer 39

(*2*1)

Answer 40

So, we literally just divided 10! by 2 factorial, right?

Answer 41

I think so. Like do you meant we divided them before we found out what to multiply?

Answer 42

Well, we multiplied down to 3 (starting from 10)

Answer 43

We didn't multiply by 2 and by 1, because those were the ninth and tenth digits, and we wer eonly looking for the first eight.

Answer 44

Oh I see, that helps me understand it a bit more. So does this "rule" follow for both permutations and combinations?

Answer 45

We can represent this mathematically.

Answer 46

In general, if you have n cases to choose from, and choose r different cases

Answer 47

nPr is shorthand for writing the permuation

Answer 48

It's shorthand, but we actually express nPr as n!/(n-r)!

Answer 49

the n! in my example would be 10, and the (n-r)! would be (10-8)!, or 2!

Answer 50

Combinations are slightly harder to explain.

Answer 51

Oh, I see. I'd imagine Combinations would be harder to explain. Because I'm not good at combinations at all haha. But I understand permutations a bit more now.

Answer 52

That's good. My head is feeling a little funny, and I'm sort of sleepy, so I'll just tell you the formulas,and how I remember them, as well as when to use which.

Answer 53

Alright, thank you so much.

Answer 54

nPr=n!/(n-r)!
nCr=nPr/r!
Permutation: order matters - common examples -> lottery ticket, password, combination lock (yes, order matters on these, despite the confusing name)
Combination: order does not matter - common exmaples -> selecting a group of people out of a larger group, putting stuff in a sandwich (lettuce and tomatos is the same as tomatoes and lettuce)

Answer 55

If you need more help, just ask more questions on Open study.

Answer 56

This was awesome Ink, seriously.

Answer 57

Thank you!

Answer 58

You're welcome :)

Answer 59

The only difference formula wise of combinations and permuations is that nCr=nPr/r!

Answer 60

Alright, I understand them a bit better thanks to you :) thank you for taking your time to help me with something I should have already got down. haha

Answer 61

Sorry for wrong answer,
Lost connection right after that.
9*8*7/3!

Answer 62

it's alright :)