Question

How many ways are there to choose 8 coins from a piggy bank containing 100 identical pennies and 80 identical nickels.
Anybody help me, please

Answer 1

i recall a formula; something like:

n!/(r1! r2! r3! ...)

Answer 2

but thats how many ways you can pick 8 coins ....

Answer 3

I have to use generating function to solve the problem. but my answer is incorrect compared to the book.. how can i get 9 with a bunch of factorial

Answer 4

using "star and bar" method leads to worse result. I stuck.

Answer 5

pppppppp

pppppppn
ppppppnp
.....

this grouping pattern follows after a row in pascals triangle

1 8 28 56 70 56 28 8 1

but each group of "k" is counted the same; this leads me to:

1 1 1 1 1 1 1 1 1 distinct groups; or 9 groups

Answer 6

recall that the sets: {a,b,c} and {b,c,a} are different permutations, but are the same combinations

Answer 7

so?

Answer 8

youll have to give me something more to go on than "so?" ....

Answer 9

sorry, I cannot see any link between them. in (a +b)^8 you have 9 terms. but the way we count is combination between 180 identical "things"

Answer 10

my question is why they have to separate pennies and nickels when both them can be counted as "coin"

Answer 11

you are being as how many different ways there are to create a group of 8 knowing that we have alot of pennies and nickels.

im not sre how the math works it but:

there is 1 way to collect 8 pennies and 0 nickels
there is 1 way to collect 7 pennies and 1 nickels
there is 1 way to collect 6 pennies and 2 nickels
there is 1 way to collect 5 pennies and 3 nickels
.... etc

Answer 12

oh oh, i guess, you have (nickel +pennies)^8 and get how many combination between them

Answer 13

there is alot of extra distraction information in the problem

Answer 14

with the given information we could delve into the probability of choosing a particular group and so forth; but thats just not relevant information needed for the solution :)

Answer 15

I have idea and need you help to develop it. are you willing?

Answer 16

wrong time of day :) ihave 3 class starting in 10 minutes that i need to get to. but id be happy to review your idea when i get back ... which is in about 5 hours :/

Answer 17

ok, I will write out all and you review later

Answer 18

have fun, good luck ;)

Answer 19

take a look at my attached file

Answer 20

isn't it nine?

Answer 21

i think that you are reading to much into the question; its not considering that each penny (or nickel) is a distinct element that must be counted as a separate element. It says that all the pennies are identical, and all the nickels are identical; and that there are enough elements of each type such that randomly picking out 8 coins would result in how many different groups of coins could be made?

It is not asking for probability of picking any one group over the other, its not asking you to classify one penny as different from another, or any other thing.

How many ways is there to grab 8 coins from the collection?

8p and 0n
7p and 1n
6p and 2n
5p and 3n
4p and 4n
3p and 5n
2p and 6n
1p and 7n
0p and 8n

there are therefore 9 ways in which we can obtain a set of 8 coins of varying combinations of pennies and nickels.

Answer 22

I got generating function from this problem is \[\frac{ 1 }{(1-x)^2} = \sum_{k=0}^{\infty} (\left(\begin{matrix}k+1 \\ k\end{matrix}\right)x^k\] when x1 + x2 =8 that means I have to figure out the coefficient of x^8. at that moment my k =8 replace to the Combination in the front of x^k I have 9C8 = 9C1 =9. That's is perfect result I get and it is similar to the book. my wonder is the "identical " thing is each of pennies and nickels is identical. that means penny 1 not= penny2 not=penny3 and so on. How can we stop there?

Answer 23

thats the exact opposite of what it means to be identical .....

if there are 3 bottles of water sitting on the table, same brand, same volume, same same identically same .... and i tell you that i need a bottle of water; does it matter which one you actually pick up?

The problem is simplifying the event such that penny1 is identical in properties to penny 2, or to penny k; such that the specific penny you pick up is immaterial.

I do not know anything about generating functions at the moment so I cant really comment if thats correct :)

Answer 24

lets say a chicken is randomly pooping out white eggs and gold eggs; how many different ways are there for a set of 8 eggs to come out?

Answer 25

thats what im reading at least; im assuming that we arent pulling coins out and then trying to determine if we are replacing them and such ....