Step 1:

Let $X$ be the random variable denoting the weight of a male student in the college.

$X\sim N(150,15^2)$

Step 2:

To find the probability that a student weighs between 120 and 155 pounds.

To find $P(120 < X < 155)$

Step 3:

Let $Z$ be the standard normal variable

$Z=\large\frac{X-\mu}{\sigma}$

$\;\;=\large\frac{X-151}{15}$

When $X=120$

$Z=\large\frac{120-151}{15}$

$\;\;=\large\frac{-31}{15}$

$\;\;=-2.07$

When $X=155$

$Z=\large\frac{155-151}{15}$

$\;\;=\large\frac{-4}{15}$

$\;\;=0.27$

Step 4:

$P(120 < X < 155)=P(-2.07 < Z < 0.27)$

$\qquad\qquad\qquad\quad\;\;=P(-2.07 < Z < 0)+P(0 < Z < 0.27)$

$\qquad\qquad\qquad\quad\;\;=P(0 < Z < 2.07)+P(0 < Z < 0.27)$ by symmetry

$\qquad\qquad\qquad\quad\;\;=0.4808+0.1064$

$\qquad\qquad\qquad\quad\;\;=0.5872$

Step 5:

The number of male students in the college=500

$\therefore$ the number of students expected to weigh between 120 and 155 pounds =$500\times 0.5872$

$\Rightarrow 244$(approx)