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

There are 65 different page arrangements that will have 18 card slots. Total possible arrangements taking into consideration the 65 different page orders is 65 * 18! or approximately 4.16154x10^17 The least common multiple of 2 and 3 is 6. And 18 divided by 6 is 3. So you'll have 3 sets of pages and each set of pages will be either 2 pages of 3 cards or 3 pages of 2 cards. So they are (3,0) = 6 pages of 3 cards. (2,1) = 4 pages of 3 cards, 3 pages of 2 cards. (1,2) = 2 pages of 3 cards, 6 pages of 2 cards. (0,3) = 9 pages of 2 cards. For each of the 4 different combinations of pages, there are multiple ways of arranging them. They are 6 pages of 3 cards, has only 1 possible arrangement. 4 pages of 3 cards, 3 pages of 2 cards, has 7!/(4!3!) = 35 2 pages of 3 cards, 6 pages of 2 cards, has 8!/(2!6!) = 28 9 pages of 2 cards, has 1 possible arrangement. So the number of different ways to have 2 card and 3 card pages is 1 + 35 + 28 + 1 = 65 different arrangements. And finally for each of the 65 different page arrangements, there are 18! different ways to order the cards to place on those pages. To demonstrate, for the first card, you can select any of the 18 cards. The second card can be any of the 17 remaining cards, the third card can be any of the 16 remaining cards, and so on. For a total of 18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 different ways to order the 18 cards. Multiply that by the 65 different page arrangements and you have 65*18! different possible arrangements.