17 students are present in $a$ class. In how many ways, can they be made to stand in 2 circles of 8 and 9 students?

Step 1 (Choose students for circles): We want to arrange 17 students into 2 distinct circles of 8 and 9 students. First, we choose 8 students from 17 to stand in the first circle:

The remaining 9 students are automatically placed in the second circle.

Step 2 (Arrange students in each circle):

• The number of ways to arrange 8 students in a circle is $(8-1)!=7!$.

• The number of ways to arrange 9 students in a circle is $(9-1)!=8!$.

Step 3 (Calculate the product): Total ways = (Ways to choose) $\times$ (Circle 1 arrangements) $\times$ (Circle 2 arrangements):

Since $9!=9\times 8\times 7!$, we simplify the fraction:

Step 4 (Conclusion): The number of ways to arrange them is exactly 17! / 72.