Answer: 22,084,920 different clubs
Step-by-step explanation:
The club must have 6 juniors and 8 seniors
We have a total of 13 juniors and 16 seniors.
Now, we know that the possible combinations of N objects into a group of K is equal to:
![C = \frac{N!}{(N-K)!*K!}](https://tex.z-dn.net/?f=C%20%3D%20%5Cfrac%7BN%21%7D%7B%28N-K%29%21%2AK%21%7D)
For the juniors we have N = 13 and K = 6
![Cj = \frac{13!}{7!*6!} = \frac{13*12*11*10*9*8}{6*5*4*3*2*1} = 1716](https://tex.z-dn.net/?f=Cj%20%3D%20%5Cfrac%7B13%21%7D%7B7%21%2A6%21%7D%20%3D%20%5Cfrac%7B13%2A12%2A11%2A10%2A9%2A8%7D%7B6%2A5%2A4%2A3%2A2%2A1%7D%20%3D%201716)
For the seniors we have N = 16 and K = 8
![Cs = \frac{16!}{8!8!} = \frac{16*15*14*13*12*11*10*9}{8*7*6*5*4*3*2*1} = 12870](https://tex.z-dn.net/?f=Cs%20%3D%20%5Cfrac%7B16%21%7D%7B8%218%21%7D%20%20%3D%20%5Cfrac%7B16%2A15%2A14%2A13%2A12%2A11%2A10%2A9%7D%7B8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%7D%20%3D%2012870)
Now, as the group consist on both combinations togheter, the number of different clubs that can be formed are:
C = Cj*Cs = 1,716*12,870 = 22,084,920