You want to arrange the books on your bookshelf by color. How many different ways could you arrange 12 books if 4 of them have a blue cover, 3 are yellow, and 5 are white?

Here we're tasked with arranging the books on a bookshelf by color of their cover but we have 12 total books. 4 of them have a blue cover, 3 of them have a yellow cover, and 5 of them have a white cover. Now this tells us that we're dealing with non distinct objects because we have multiple objects of the same color. They're non distinguishable or non distinct. So here, we're going to use our formula for permutations of non distinct objects. Now the total number of objects we have is 12, so this represents our value for n. Then we have 3 other different types of things. So we have 4 blue objects, 3 yellow, and 5 white. So this represents our 3 different values for r. We can consider this r 1, r 2, and r 3. Now plugging this into our permutations for non distinct objects formula, we end up with 12 factorial divided by 4 factorial times 3 factorial times 5 factorial. Now from here, we want to go ahead and cancel the largest factorial on the bottom, which in this case is 5 factorial. So we're going to rewrite our numerator in order to cancel that 5 factorial. So let's go ahead and do that. So our numerator can be rewritten as 12 times 11 times 10 times 9 times 8 times 7 times 6 finally times 5 factorial. Sometimes you're just going to have to really expand out your numerator, but that's okay. Now this gets divided by 4 factorial times 3 factorial times 5 factorial. But we know that that 5 factorial cancels. So I can go ahead and cancel that out and then I'm going to rewrite my expanded numerator and then go ahead and expand my denominator as well. So my numerator remains is still 12 times 11 times 10 times 9 times 8 times 7 times 6 and then expanding out my denominator, 4 factorial expands out to 4 times 3 times 2 times 1 and that's all getting multiplied by 3 factorial, which is 3 times 2 times 1. Now going ahead and multiplying everything out here in my numerator, I end up getting a really high number, 3,000,000 991,680, and then my denominator is a bit smaller. It's just 144. Now fully dividing that out, I end up getting 27,720 which is a ton of different ways that we could organize these 12 books on that shelf. Thanks for watching, and let me know if you have questions.