Roberto wants to display his 18 sports cards in an album. some pages hold 2 cards and others hold 3 cards. how many different ways can roberto display his figures?

Let x = number of pages holding 2 cards,
y = number of pages holding 3 cards.

So his total number of card would be 2x + 3y, which is given as 18. So we get this equation.

2x + 3y = 18

Subtract 3y from both sides:

2x  = 18 – 3y

Divide both sides by 2:

x = (18 – 3y)/2

x must be an integer. So 18–3y must be a multiple of 2. Since 18 is a multiple of 2, 3y is also a multiple of 2. Then y must be a multiple of 2 too.

Let's start with y = 0.
When y = 0, x = (18 – 3(0))/2 = 9.
When y = 2, x = (18 – 3(2))/2 = 6.
When y = 4, x = (18 – 3(4))/2 = 3.
When y = 6, x = (18 – 3(6))/2 = 0.

Thus there are 4 ways to display his figures.