5 cards from a 52 card deck, what is the probability that the sum of cards is greater than 48 The problem states: 
$5$ cards are dealt from a standard $52$ card deck. What is the probability that the sum of the values on the five cards is $48$ or more?
It is assumed of course that the value of face cards is $10$ and that of aces $11$. I know I am looking for the ratio between the number of possible outcomes with sum of values at least $48$, and the total number of possible outcomes, but I am having trouble finding the former quantity.
Any help is appreciated, thank you!
 A: The number of possible hands, even treating every card as unique, is only $\binom{52}{5} = 2598960$, which is well within brute-force range of a computer.
It's possible to do better, i.e. jointly polynomial in the deck size, hand size, and range of card values, using a dynamic programming algorithm. We can find the distribution of the sum of 5 cards drawn from the deck by iterating over the number of aces $i = 0 \ldots 5$ we could potentially draw:

*

*Recursively compute the distribution of the sum of $5 - i$ cards drawn from a deck without aces, memoizing the solution.

*Add the value of the $i$ aces.

and then binomially weighting the contributions for each $k$. The recursive calls then compute the sum of $0 \ldots 5$ cards drawn from a deck with the 10s/faces also removed, then the 9s also removed, and so forth until only 2s are left. This corresponds to the decomposition of the multivariate hypergeometric distribution as a product of binomials.
I've implemented this in my Icepool Python library. You can run this script online:
from icepool import Deck

deck = Deck([11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 10], times=4)
output(deck.deal(5).sum())

The result is that 55580 out of 2598960 possible hands sum to at least 49, or about 2.14%. Compare 2.60% with replacement (e.g. dice rather than cards).
This approach can be extended to find X-of-a-kind, straights, and more. If you are curious to learn more, you can read my paper on the subject.
@inproceedings{liu2022icepool,
    title={Icepool: Efficient Computation of Dice Pool Probabilities},
    author={Albert Julius Liu},
    booktitle={Eighteenth AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment},
    volume={18},
    number={1},
    pages={258-265},
    year={2022},
    month={Oct.},
    eventdate={2022-10-24/2022-10-28},
    venue={Pomona, California},
    url={https://ojs.aaai.org/index.php/AIIDE/article/view/21971},
    doi={10.1609/aiide.v18i1.21971}
}

