While multiplying four real numbers, John took one of the numbers as 6

Based on the question, we can form the equation 26XYZ+720= 62XYZ , considering X,Y,Z are the the other 3 numbers.

Therefore, the equation reduces to 36XYZ=720 or XYZ=20.

To get the minimum sum of the cubes we assume we assume that X=Y=Z , and in turn, this means X=Y=Z= cube root (20).

Which implies, the minimum sum of the cubes of X,Y & Z is 60.

You can take four nos. as a,b,c and 26 and their product will be x
abc26=x
we can rewrite it as abc=x/26 -----------(1)

If digits of 26 are reversed then their product increases by 720-
so we can write that by- abc62=x+720
Now lets put in the value of abc from equation (1) here

x/26 * 62=x+720
x=520
so, abc=20

Now we have to select the real nos., the number will be an integer as the sum of cube of those nos. will be an integer.
There are 2 cases-(not taking + & - ones)
5*2*2 and 5*4*1
To make the answer choice possible 2 digits have to be negative (as if we take all the digits as positive the answer will not match)

Case 1) 5*2*2
Let's take (-5)^3+2^3+(-2)^3=-125, not matching
5^3+(-2)^3+(-2)^3=+125, not matching
so this case is not possible.

Case2) 5*4*1
(-5)^3+4^3+(-1)^3=-62, not matching
5^3+(-4)^3+(-1)^3= 60, This answer is matching the answer choice C
So this is the correct answer.