1) At a college library exhibition of faculty publications, 3 math books, 4 social science books, and 3 bio books will be displayed on a shelf (assume that all books are different.)
In how many ways can the ten books be arranged on the shelf if books on the same subject matter are placed together?
(What I did)
For math: P(3,3)
For s.s: P(4,4)
For bio: P(3,3)
math and bio : P(3,3)=6
s.s: P(4,4)=24
I multiplied 6*6*24=864
correct
To calculate the total number of ways the books can be arranged on the shelf, we need to consider the arrangements for each subject separately and then multiply them together.
For the math books: There are 3 math books, and we want to arrange them among themselves. This can be done in P(3,3) ways, which is equal to 3!.
For the social science books: There are 4 social science books, and we want to arrange them among themselves. This can be done in P(4,4) ways, which is equal to 4!.
For the bio books: There are 3 bio books, and we want to arrange them among themselves. This can be done in P(3,3) ways, which is equal to 3!.
Now, since the math books, social science books, and bio books need to be placed together, we consider them as individual entities. Therefore, the three subject groups can be arranged among themselves in P(3,3) ways.
To find the total number of arrangements, we multiply the arrangements for each subject with the arrangement of the subject groups:
P(3,3) * P(4,4) * P(3,3) * P(3,3) = 6 * 24 * 6 * 6 = 5,184
So, there are 5,184 ways in which the ten books can be arranged on the shelf if books on the same subject matter are placed together.
To find the number of ways to arrange the ten books on the shelf while keeping books of the same subject matter together, you correctly calculated the number of ways to arrange each subject's books separately:
For the 3 math books, you found the number of permutations using P(3,3) = 3! = 6.
For the 4 social science books, you found the number of permutations using P(4,4) = 4! = 24.
For the 3 bio books, you also found the number of permutations using P(3,3) = 3! = 6.
To find the total number of ways to arrange all the books while keeping books of the same subject together, you should multiply these individual counts since the arrangement of each subject's books is independent of the others:
6 (math) * 6 (bio) * 24 (social science) = 864.
So there are a total of 864 ways to arrange the ten books on the shelf while keeping the books of the same subject matter together.