The number of ways of choosing  8 students out of 17 students to stand in a circle of 8 is same as the number of ways of choosing 9 students out of 17 students to stand in another circle of 9 students.
That is 17C 8= 17 C9
The  number of ways  of choosing 8 students out of 17 students is 17 C8= 17!/( 8!*9!)= (17*16*15*14*13*12*11*10)/(1*2*3*4*5*6*7*8)=24310
Once you choose 8 students the other 9 students will be automatically selected.
So total number of ways of choosing  17 students in 2 circles of 8 and 9 students is 23,410.