$S = \{1,2,3,\ldots,4999,5000\}$
Let 'A' be the set of integers divisible by $3 \implies A =\{3,6,9,12,15,\ldots,4998\}$
Let 'B' be the set of integers divisible by $4 \implies B= \{4,8,12,16,\ldots,5000\}$
Let 'C' be the set of integers divisible by both $3$ and $4 \implies C = \{12,24,36,\ldots,4992\}$
$|A| = 3+(n-1)3 = 4998 \implies n = 4998/3 = 1666$
$|B| = 4+(n-1)4 = 5000 \implies n = 5000/4 = 1250$
$|C| =12+(n-1)12 = 4992 \implies n = 4992/12 = 416$
Divisible by $3$ or $4:$ $\{3,4,6,8\dots 5000\}$ : $|D| = |A| + |B| -|C|$
$= 1666+1250-416$
$= 2500$
Number of integers that is divisible by neither $3$ nor $4$
$= |S| - |D|$
$= 5000-2500$
$= 2500$
So, C is the correct answer.