let

x = the number of packages of 8, x>=0, x is integer

y = the number of packages of 24, y>=0, y is integer

8x + 24y = 48 divide both sides by 8

x + 3y = 6

x = 6 - 3y

x >= 0

6 - 3y >= 0

y <= 6/3

y <= 2

for y = 0 we have x = 6 follows you can buy 48 cookies by buying 6 packages of 8 cookies.

for y = 1 we have x = 6 - 3*1 = 3 follows you can buy 48 cookies by buying 3 packages of 8 cookies and 1 package of 24 cookies.

for y = 2 we have x = 6 - 2*3 = 0 follows you can buy 48 cookies by buying 2 packages of 24 cookies.

You can by 48 cookies in 3 possbile ways.