The three

The three boys, Adam, Boris, and Cyril, will be taken on a two-seater ski lift. How many different removal options are there? What would it be like if four boys or five were to be taken away?

Final Answer:

n₃ =  3
n₄ =  6
n₅ =  60

Step-by-step explanation:

$$\begin{gathered} C_2(3)=\left(\genfrac{}{}{0ex}{}{3}{2}\right)=\frac{3!}{2!(3-2)!}=\frac{3}{1}=3 \\ {n}_3=\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3 \end{gathered}$$

$$\begin{gathered} C_2(4)=\left(\genfrac{}{}{0ex}{}{4}{2}\right)=\frac{4!}{2!(4-2)!}=\frac{4\cdot 3}{2\cdot 1}=6 \\ {C}_2(2)=\left(\genfrac{}{}{0ex}{}{2}{2}\right)=\frac{2!}{2!(2-2)!}=\frac{1}{1}=1 \\ {n}_4=\left(\genfrac{}{}{0ex}{}{4}{2}\right)\cdot \left(\genfrac{}{}{0ex}{}{2}{2}\right)=6\cdot 1=6 \end{gathered}$$

$$\begin{gathered} C_1(2)=\left(\genfrac{}{}{0ex}{}{2}{1}\right)=\frac{2!}{1!(2-1)!}=\frac{2}{1}=2 \\ {C}_2(5)=\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5\cdot 4}{2\cdot 1}=10 \\ {n}_5=\left(\genfrac{}{}{0ex}{}{5}{2}\right)\cdot \left(\genfrac{}{}{0ex}{}{5-2}{2}\right)\cdot \left(\genfrac{}{}{0ex}{}{5-2-2}{2}\right)=10\cdot \left(\genfrac{}{}{0ex}{}{5-2}{2}\right)\cdot \left(\genfrac{}{}{0ex}{}{5-2-2}{2}\right)=60 \end{gathered}$$

Try another example