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Sagot :
We have the following:
- 5 possible about genetics and need fewer than 3 (so it can be 0, 1 or 2).
- 8 about ethics
- want to select 6 in total.
We can calculate all the possible ways by doing it in three situations:
1 - From the 6, 0 will be genetics and 6 will be ethics
2 - From the 6, 1 will be genetics and 5 will be ethics
3 - From the 6, 2 will be genetics and 4 will be ethics
All of these will have to add up to find the total number of ways.
1 - 0 genetics, 6 ethics:
Since no genetics will be chosen, we can choose any 6 from the 8 possible about ethics, that is, we have a situation of "8 choose 6"
The equation for a situation "n choose k" and the number of ways in it is:
[tex]n=\frac{n!}{k!(n-k)!}[/tex]So, if we have "8 choose 6":
[tex]n_1=\frac{8!}{6!(8-6)!}=\frac{8\cdot7\cdot6!}{6!2!}=\frac{8\cdot7}{2}=4\cdot7=28[/tex]So, in this first we have 28 ways.
2 - 1 genetics, 5 ethics:
Here, we will have one equation for each and the total number of ways will be the multiplication of both.
For genetics, we have to pick 1 from 5, so "5 choose 1":
[tex]\frac{5!}{1!(5-1)!}=\frac{5\cdot4!}{4!}=5_{}[/tex]For ethics, we have to pick 5 from 8, so "8 choose 5":
[tex]\frac{8!}{5!(8-5)!}=\frac{8\cdot7\cdot6\cdot5!}{5!3!}=\frac{8\cdot7\cdot6}{3\cdot2}=8\cdot7=56[/tex]So, the total number of ways is the multiplicatinos of them:
[tex]n_2=5\cdot56=280[/tex]3 - 2 genetics, 4 ethics:
Similar to the last one.
For genetics, we have to pick 2 from 5, so "5 choose 2":
[tex]\frac{5!}{2!(5-2)!}=\frac{5\cdot4\cdot3!}{2\cdot3!}=\frac{5_{}\cdot4}{2}=5\cdot2=10[/tex]For ethics, we have to pick 4 from 8, so "8 choose 4":
[tex]\frac{8!}{4!(8-4)!}=\frac{8\cdot7\cdot6\cdot5\cdot4!}{4!4!}=\frac{8\cdot7\cdot6\cdot5}{4\cdot3\cdot2}=2\cdot7\cdot5=70[/tex]So, the total number of ways is the multiplicatinos of them:
[tex]n_3=10\cdot70=700[/tex]Now, the total number of ways is the sum of all these possibilities, so:
[tex]\begin{gathered} n=n_1+n_2+n_3 \\ n=28+280+700 \\ n=1008 \end{gathered}[/tex]So, the total number of ways is 1008.
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