Answer to Question #194633 in Combinatorics | Number Theory for Ayanda M

Question

 Polya described four steps in the solving of a mathematics problem. Here is a problem: A classroom has 2 rows, each with 8 seats. Of 14 students, 5 always sit in the front row, and 4 always sit in the back row, and the rest sit in either row. In how many ways can the students be seated?

6.1.1 Use three different types of representations to model the problem

6.1.2 Now use the steps of Polya to solve this problem. You must explain in detailwhat you are doing in each step

Answer 1

6.1.1

5 students can be seated in the first row in $^8P_5$ ways.

4 students can be seated in the second row in $^8P_4$ ways.

Remaining 5 students can be seated in remaining 7 seats in $^7P_5$ ways.

6.2.2

Total number of ways students can be seated is-

$=^8P_5\times ^8P_4\times ^7P_5 \\[9pt] =\dfrac{8!}{3!}\times \dfrac{8!}{4!}\times \dfrac{7!}{2!} \\[9pt] =(8!)^2\times \dfrac{5.7}{2} \\[9pt] =\dfrac{35}{2}(8!)^2$