Footballers 2

Footballers have jerseys numbered 7, 8, 9, 10, and 11. The coach wants to send them to attack:
a) so that no two even-numbered jerseys are adjacent,
b) so that no two odd-numbered jerseys are adjacent.

How many options does he have in each case?

Final Answer:

a =  72
b =  12

Step-by-step explanation:

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

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

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