Question

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?

Answer 1

Let x denote the number of pages holding 2 cards and y denote the number of pages holding 3 cards.

So his entire number of card would be represented by 2x + 3y, which is given as 18.

2x + 3y = 18

Subtract 3y from the two sides:

2x = 18 – 3y

then divide both sides by 2:

x = (18 – 3y)/2

We know that 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.