Assume the cards have the following values:
K, Q, J, = 10
Ace-10 = 1-10
The long way to do this would be to calculate ways the cards can add up to 20 and divide by $50C4$ and do this for every possible way. For example:
Assume I draw a 2 of hearts, Jack of Spades, 5 of clubs and a 3 of diamonds.
That's one possible combination, $\left(\frac{4C4}{52C4}\right)$ You'd then have to find all the other possible combinations and add them together.
To find all the possible combinations this way would be too exhausting and time consuming, so I was wondering if there was another way to approach this question
Iterating through all the possible four-card hands with a computer program (I used the Python itertools module) shows that the number of hands which sum to 20 is $10,225$. So the probability that the cards will sum to 20 is $$\frac{10,225}{\binom{52}{4}}=0.0377690$$