Sets and Counting

Question

Suppose there are three pitchers and three catchers at a baseball training camp.

In how many ways can the six players be seated in a row if no one sits next to someone who plays the same position?

Answer 1

C1, C2, C3, P1, P2, P3

suppose a pitcher sits first

how many ways can the pitchers be seated ?

P1    P2    P3 with the catchers in between the pitchers

P1    P3    P2

P2    P1    P3

P2    P3    P1

P3    P1    P2

P3    P2    P1

there are 6 ways the pitchers can be seated

for each of these 6 ways, the catchers can be seated in six ways

the catchers are between the pitchers

P1 C1 P2 C2 P3 C3

P1 C1 P2 C3 P3 C2

P1 C2 P2 C1 P3 C3

P1 C2 P2 C3 P3 C1

P1 C3 P2 C1 P3 C2

P1 C3 P2 C2 P3 C1

now do the same for the other 5 arrangements of pitchers

so there are 6*6=36 arrangements with a pitcher sitting first

now do the same thing but have a catcher sit first

you'll have 6 arrangements of catchers

for each arrangement of catchers there will be 6 arrangement of pitchers as shown above

so there will be 6*6=36 arrangements with a catcher sitting first

36+36=72 different arrangements