In how may ways can 15 identical books of mathematics and 12 identical books of physics be placed in a row on a shelf so that tow books of physics may not be together ?

Explanation

In order that no two books of physics are togethor, we must first arranged all books of mathematics in a row, since all books of mathematics are identical , so they can be arranged in a row in only onwe way as shown below:
`"_ "M"_ M _ M _ "............."_ M _ M _"`
Here, M denotes the position of a mathematics book are _ that of a physics book. Since, there are 15 books of mathematics, the number places mark _ are therefore 16. Now, 12 books of physics are to be arranged in these 16 places so that no two of them are together. This can be done in `""^(16)C_(12)` ways.
Since all books of physics are identical, so 12 books of physics can be arranged in 16 places in only one way.
`therefore ` Required number of ways`=1xx""^(16)C_(12)=1820`