HCF of $n$ and $240$ is $1$
$240 = 2^4 \times3\times 5$
so we need to find the numbers, such that, $120 \leq n \leq 240$ , and not divisible of $2$ or $3$ or $5$
Numbers divisible by $2$ such that $120 \leq n \leq 240 = 120 - 59 = 61$
Numbers divisible by $3$ such that $120 \leq n \leq 240 = 80 -39 = 41$
Numbers divisible by $5$ such that $120 \leq n \leq 240 = 48 - 23 = 25$
Numbers divisible by $2$ and $3$ such that $120 \leq n \leq 240 = 40 -19 = 21$
Numbers divisible by $2$ and $5$ such that $120 \leq n \leq 240 = 24 -11 = 13$
Numbers divisible by $3$ and $5$ such that $120 \leq n \leq 240 = 16 -7 = 9$
Numbers divisible by $2$ ,$3$,and $5$ such that $120 \leq n \leq 240 = 8 -3 = 5$
Numbers divisible by $2$ or $3$ or $5$ such that $120 \leq n \leq 240 = 61+41+25-21-13-9+5= 89$
Total numbers such that $120 \leq n \leq 240 =121$
Total numbers such that $120 \leq n \leq 240$ and not divisible by $2$ or $3$ or $5 = 121-89 = 32$