Footballers 2

Footballers have jerseys with numbers 7, 8, 9, 10, 11. The coach wants to send them to attack
a) so that even jersey numbers are not next to each other
b) so that odd jersey numbers are not next to each other.

How many options does he have?

Correct answer:

a =  72
b =  12

Step-by-step explanation:

$$\begin{gathered} 7,8,9,10,11 \\ {A}_1:7,9,11 \\ {A}_2:xNxNxNx \\ {C}_2(4)=\left(\genfrac{}{}{0ex}{}{4}{2}\right)=\frac{4!}{2!(4-2)!}=\frac{4\cdot 3}{2\cdot 1}=6 \\ a=3!\cdot \left(\genfrac{}{}{0ex}{}{4}{2}\right)\cdot 2!=3!\cdot 6\cdot 2!=72 \end{gathered}$$

$$\begin{gathered} B_1:8,10 \\ {B}_2:xPxPx \\ {C}_3(3)=\left(\genfrac{}{}{0ex}{}{3}{3}\right)=\frac{3!}{3!(3-3)!}=\frac{1}{1}=1 \\ b=2!\cdot \left(\genfrac{}{}{0ex}{}{3}{3}\right)\cdot 3!=2!\cdot 1\cdot 3!=12 \end{gathered}$$

Try another example