In how many ways can 10 H.S. students and 7 B.Com. students be arranged in a line so that no two B.Com. students may sit together?

Q.  In how many ways can 10 H.S. students and 7 B.Com. students be arranged in a line so that no two B.Com. students may sit together?

 Solution Clearly, 10 H.S. students can be arranged among themselves in (10)! ways. For each of these (10)!  arrangements of H.S. students, no two B.Com. students willc pcupy their seats in consecutive positions if they are placed in between or at the two extremes of the H.S. students. Clearly, there are 11 such places where 9 places in between and 2p laces at the two extremes; where B.Com. students can be placed as follows

[H,"H,°H,"H,°"H,°H,"H,°H,°H,°H0

Here, H,H,HH10 represents a particular arrangement of 10 H.S. students and B.Com. students are to be placed in places shown by".]

Now, 7 B.Com. students can be placed in these 11 places in P¹¹ ways.

Therefore, the required number of arrangements=(10)! x"P7