Question

On planet Enigma, the residents use a currency called the confusion. There are only 2 confusion bills on Enigma, one worth 8 confusions and the other worth 11 confusions. There are also some coins of smaller value, but each weighs over 10 kilograms, so they are difficult to carry around.

In how many ways can a resident of Enigma use only bills to purchase a toaster that costs 96 confusions?

Answer 1

(1 + x^8 + x^16 + x^24 + ... + x^96)(1 + x^11 + x^22 + ... + x^88) =

Stuff + 2x^96 + Stuff

Thus, 2

Answer 2

@tkhunny not sure if I understand from this, can you elaborate pl.

Answer 3

We're just using Generating Functions. Have you not encountered this in your Discrete Mathematics?

Answer 4

How many ways are there to make 1¢-12¢ from Pennies, Nickles and Dimes?

(1 + x + x^2 + ... + x^14) -- The pennies generator. Exponent increases by 1 each time.
(1 + x^5 + x^10) -- The nickles generator. Exponent increases by 5 each time.
(1 + x^10) -- The dimes generator. Exponent increases by 10 each time.

(1 + x + x^2 + ... + x^14)(1 + x^5 + x^10)(1 + x^10) =

(1 + x + x^2 + x^3 + x^4) + 2(x^5 + x^6 + x^7 + x^8 + x^9) + 4(x^10 + x^11 + x^12) + Other Stuff

Thus,
1 way to make 1¢-4¢ <== Makes sense, since you can use only pennies.
2 ways to make 5¢-9¢ <== Makes sense, since you can swap out 5 pennies for a nickle.
4 ways to make 10¢-12¢

It's a powerful tool, Generating Functions.

Answer 5

thanks!