ganeshie8
  • ganeshie8
How many total ways are there to make one dollar ?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
schrodinger
  • schrodinger
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ganeshie8
  • ganeshie8
|dw:1449945419248:dw|
ganeshie8
  • ganeshie8
using above denominations...
anonymous
  • anonymous
29 in coins but with a dollar bill 30

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More answers

ganeshie8
  • ganeshie8
100 is a very good guess, but it isn't correct...
anonymous
  • anonymous
wait it is 293
ganeshie8
  • ganeshie8
293 is correct !
Christos
  • Christos
how did you find ?
ganeshie8
  • ganeshie8
Could you also show how you arrived at that :)
anonymous
  • anonymous
you have to figure out how many half dollar coins you need how many pennies you need and how many nickels you need
anonymous
  • anonymous
but the cheating way is looking it up
ParthKohli
  • ParthKohli
oh I know this\[1x_1+5x_2+10x_3+25x_4+50x_5+100x_6=100\]nonnegative integral solutions of this
anonymous
  • anonymous
what the frik are those
malcolmmcswain
  • malcolmmcswain
http://prntscr.com/9ddfxx
malcolmmcswain
  • malcolmmcswain
:/
ParthKohli
  • ParthKohli
now what we do is apply a trick to convert this to an algebra problem\[(1+x+x^2+\cdots)(1+x^5+x^ {10}+\cdots)(1+x^{10}+{x}^{20}+\cdots)\cdots\]consider this expression. we can find the coefficient of \(x^{100}\) and that is our answer.
Christos
  • Christos
I would image solving this with recursion in programming . Wouldn't you otherwise had to build a huge tree of every combination by hand
ganeshie8
  • ganeshie8
generating functions make the life easy but it seems finding the coefficient isn't easy here...
ParthKohli
  • ParthKohli
wait do we only have the coins given in the picture?
ParthKohli
  • ParthKohli
am I allowed to use a cent hundred times? because that picture only shows two of them
ganeshie8
  • ganeshie8
Ofcourse coins can be used any number of times
ParthKohli
  • ParthKohli
thanks.\[\frac{1}{1-x}\cdot\frac{1}{1-x^5}\cdot \frac{1}{1-x^{10}}\cdots \frac{1}{1-x^{100}}\]
ParthKohli
  • ParthKohli
lol yeah, finding the coefficient is tough but not impossible
ganeshie8
  • ganeshie8
http://www.wolframalpha.com/input/?i=maclaurin+series+of+1%2F%28%281-x%29%281-x%5E5%29%281-x%5E10%29%281-x%5E25%29%281-x%5E50%29%281-x%5E100%29%29
ganeshie8
  • ganeshie8
keep clicking "More terms..."
ParthKohli
  • ParthKohli
nice pattern. 1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 50...
Christos
  • Christos
int make_change(int amount, int[] denominations) { int[] change = new int[amount+1]; change[0] = 1; for (int i = 1; i <= amount; i++) { for (int denomination : denominations) { if (i >= denomination) { change[i] += change[i-denomination]; } } } return change[denomination]; }
ParthKohli
  • ParthKohli
well at least we could convert this problem to a form which Wolfram|Alpha accepts haha
ganeshie8
  • ganeshie8
Nice :)
ParthKohli
  • ParthKohli
what is your solution?
ganeshie8
  • ganeshie8
Page11 example 3.5 http://db.math.ust.hk/notes_download/elementary/algebra/ae_A11.pdf
ganeshie8
  • ganeshie8
that solution is a bit complicated..
ParthKohli
  • ParthKohli
oh haha but he did use generating functions
ikram002p
  • ikram002p
nice! i like those generating function problems. i have worked on generating integrals like Voltera equations very interesting ways to solve :O
danica518
  • danica518
start with 1 cent splits
danica518
  • danica518
|dw:1449953244057:dw|
danica518
  • danica518
|dw:1449953553432:dw|
danica518
  • danica518
how about building up the restrictictions like so, the way to use the lower numbers is they will always sum to a coin of a higher value
danica518
  • danica518
for example if a 5 is used, there there have to k number of sets of 1s that are a multiple of 5, if 1 cent is used
ganeshie8
  • ganeshie8
yeah the coefficients in generating function are also changing in multiples of 5 http://www.wolframalpha.com/input/?i=maclaurin+series+of+1%2F%28%281-x%29%281-x%5E5%29%281-x%5E10%29%281-x%5E25%29%281-x%5E50%29%281-x%5E100%29%29

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