Min-coin change problem with limited coins

I have been assigned the min-coin change problem for homework. I have to calculate the least number of coins needed to make change for a certain amount of cents in 2 scenarios: we have an infinite supply of coins and also where we only have 1 of each coin. I was able to quickly implement the solution for the first scenario using a recursive formula from here but was not able to find a recursive formula for the second case and I instead came up with an iterative solution.

We can just sum every possible combination of coins, check and see if the sum is equal to the target wanted and compare the number of coins used to the current lowest number of coins. I also realized that I can use binary counting to simulate all possible combinations using the bits in the iterator as a coefficient (either 1 or 0) for each coin and then calculate the Hamming Weight to check how many coins was used in the sum.

To visualize an example with 3 coins:

C1 C2 C3 | S                        | W  0  0  0 | 0 * C1 + 0 * C2 + 0 * C3 | 0  0  0  1 | 0 * C1 + 0 * C2 + 1 * C3 | 1  0  1  0 | 0 * C1 + 1 * C2 + 0 * C3 | 1  0  1  1 | 0 * C1 + 1 * C2 + 1 * C3 | 2  1  0  0 | 1 * C1 + 0 * C2 + 0 * C3 | 1  1  0  1 | 1 * C1 + 0 * C2 + 1 * C3 | 2  1  1  0 | 1 * C1 + 1 * C2 + 0 * C3 | 2  1  1  1 | 1 * C1 + 1 * C2 + 1 * C3 | 3 

Obviously we can omit the case where all the coefficients are 0, but for completeness (when target = 0) and simplicity I’ve kept it in the solution.

My implementation in Python is:

import math   def least_coins(coins, target):     least = math.inf      # loop all 2^n possible sums of coins     for i in range(2 ** len(coins)):         sum = 0          # add all the coins multiplied by their coefficient in this iteration         for j in range(len(coins)):             # get the j-th bit in i             coefficient = i >> j & 1             sum += coefficient * coins[j]          if sum == target:             count = 0             # calculate the Hamming Weight of i to determine the number of coins used             while i:                 i &= i - 1                 count += 1             least = min(least, count)      return least   coins = [3, 1, 1, 3, 2, 4] target = 10 print(least_coins(coins, target)) >>> 3 # 3 + 3 + 4 = 10 

As you can see the coins are not necessarily unique, but can only be used once. How can I turn this into a recursive formula (if possible and for the reason of possibly being able to optimize it using dynamic programming) or even introduce coefficients other than 0 and 1 so the problem can be more general where we have a certain amount of each coin?

Can limited wish be used to cast spells from prohibited schools of magic?

I’m playing a silverbrow human sorcerer in D&D 3.5e and a prestige class I plan on taking in the future requires me to select a school of magic to prohibit. I’m having trouble deciding which school to choose, so I’m wondering if limited wish would allow me to duplicate the effect of a spell from the prohibited school.

D&D 5e: Can the Wish spell overcome limited magic immunity?

I always wondered how the two interacted. What if I cast Wish, copying the spell Contagion (5th level), at a rakshasa? The creature is immune to spells of 6th level or under, but does my spell count as 9th level because I used wish, or 5th level because the effect is that of contagion? Perhaps it counts as 9th for counterspell purposes, but 5th level for immunity purposes? If anyone has an answer, I would love to hear it.

Can some one please explain what is meant by “5 level lower” with Kineticist Archetype Elemental Purist’s Limited Buffer feature?

Can some one please explain what is meant by “5 levels lower” with Kineticist Archetype Elemental Purist’s Limited Buffer feature?

So far I get that you just get it and it benefits later at level 11. Isn’t it?

How is repeated use of Illusory Reality limited?

I’m planning on creating an advanced Illusionist for a high level campaign. When reading through the rules for said school of wizardry, I came across this:

When you cast an illusion spell of 1st level or higher, you can choose one inanimate, nonmagical object that is part of the illusion and make that object real. You can do this on your turn as a bonus action while the spell is ongoing. The object remains real for 1 minute.

This seems ridiculously overpowered, even for a level 14+ character, given that one can combine it with the action of Minor Illusion, a cantrip, so the uses are near infinite. Is there a limit on the number of times this can be done?

For example, if a ceiling were falling, couldn’t I just cast and recast all sorts of barriers and objects to stop its descent?

Can the spells combine, or is there an unexpected consequence or something entirely different? If there isn’t, this campaign will be a breeze, which won’t be nice for a one-shot (you can’t keep treasure, so it has to be about the journey, not the reward).

Need help with showing information in a limited space

Well, it is my first time seeking help for user experience design, so pardon me as I am not accustomed to the proper rules.

I have an information card which shows a thumbnail for a video and below it, information about that video is displayed. The layout is basically like this,

enter image description here

Well, the difficulty is that the application allows user to add multiple number of fields and value, which means that along with video title and details. The user can add 10 more fields with corresponding values.

I was thinking of using a modal to display additional information about the video, is there any better way as I am not so sure about it.

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