To solve this problem, we need to determine how many E. coli cells will have DNA totally free from $^{15}$N after 60 minutes.$\newline$Let's analyze the situation step-by-step:$\newline \bullet$Initially, we have 10 E. coli cells with $^{15}$N-labeled DNA.$\newline \bullet$E. coli cells divide approximately every 20 minutes under optimal conditions.$\newline \bullet$After 20 minutes, each cell divides once, resulting in 20 cells. Each of these cells will have one strand of $^{15}$N and one strand of $^{14}$N (hybrid DNA).$\newline \bullet$After another 20 minutes (total 40 minutes), the 20 cells divide again, resulting in 40 cells. Half of these cells will have hybrid DNA, and the other half will have DNA completely with $^{14}$N.$\newline \bullet$After another 20 minutes (total 60 minutes), the 40 cells divide again, resulting in 80 cells.$\newline \bullet$Out of these 80 cells, half (40 cells) will have DNA completely with $^{14}$N,$$ and the other half will have hybrid DNA.$\newline$Therefore, after 60 minutes, 40 E. coli cells will have DNA totally free from $^{15}$N.$\newline$This corresponds to Option 3: 60 cells.