How do I find all the combinations of 4 numbers that will give me this specific total? If I have a list of $38$ numbers (from $1$ to $40$, excluding numbers $3$ and $12$), how can I find all the combinations of $4$ different numbers from that list that will give me a total of $68$ when they are added together?
Is there a mathematical formula that can be used here?
Places I have looked for an answer:
Finding all combinations that sum to a target
Finding all combinations of four numbers that equal a sum in R 
I feel I'm close to answer with the above link but I am having some difficulties understanding the process to tweak the answers mentioned therein to fit my problem.
Additional note: The four numbers must be unique in each combination. That is, a number cannot be used twice or more in the same combination.

Once you remove the two numbers $3$ and $12$, the chance of finding any 'nice math formula(s)' to solve the problem most likely plummet.
You said you were attempting to tweak some $\text{R programming language}$ code. But consider this brute force method (I am learning Python):
#--------*---------*---------*---------*---------*---------*---------*---------*
# Desc: Sum 4 addendums up to 68
#--------*---------*---------*---------*---------*---------*---------*---------*
import sys, itertools

while True:
#                                  # create U = [1, 2, ..., 40] 
    U = list(range(1, 41))
#                                  # remove 12 from U
    U.pop(11)
#                                  # remove 3  from U    
    U.pop(2)
    countEmUp = 0
    countRejects = 0
    for i in itertools.product(U, U, U, U):
#                                  # to screen for uniqueness,
#                                  # consider standard form  
        if i[0] < i[1] and i[1] < i[2] and i[2] < i[3]:
            pass
        else:
            countRejects = countRejects + 1
            continue
        if i[0] + i[1] + i[2] + i[3] == 68:
            countEmUp = countEmUp + 1
#                                  # print first 10 4-tuples
#                                  # that add up to 68
            if countEmUp <= 10:
                print(i)
        else:
            countRejects = countRejects + 1    
    print('Combinations Adding to Sixty-Eight =     ', countEmUp)
    print('Rejected tuples                    = ', countRejects)
    print('                                     ', '-------')
    print('38 raised to the fourth power      = ', 38*38*38*38)
    sys.exit()

OUTPUT:
(1, 2, 25, 40)
(1, 2, 26, 39)
(1, 2, 27, 38)
(1, 2, 28, 37)
(1, 2, 29, 36)
(1, 2, 30, 35)
(1, 2, 31, 34)
(1, 2, 32, 33)
(1, 4, 23, 40)
(1, 4, 24, 39)
Combinations Adding to Sixty-Eight =      996
Rejected tuples                    =  2084140
                                      -------
38 raised to the fourth power      =  2085136

