Question

(d) The cats and dogs, which are all distinguishable, are going to have a tug-of-war against the students. Assume there are exactly five cats and six dogs willing to participate. For the following problems calculate the exact answers by hand and show your work. i. How many teams consisting of exactly eight animals are there? ii. How many teams consisting of exactly four cats and exactly four dogs are there? iii. How many teams consisting of exactly four cats and exactly four dogs are there if there is a pair of cats that refuse to be on the team with each other? iv. How many teams consisting of exactly four cats and exactly four dogs are there if there is a pair of dogs that refuse to be on the team with each other?

Answer 1

5 cats and 6 dogs are there to participate.

C1, C2 ,C3 , C4 , C5 and D1, D2, D3, D4, D5, D6

1) Total number of teams with 8 animals =

8166

2) Total number of teams consisting of four cats and four dogs =

5 7

3) Total number of teams consisting of four cats and four dogs when two cats refuse to be in same team =

×2× = 30

4) Total number of teams consisting of four cats and four dogs when two cats refuse to be in same team = $\binom{5}{4}\times2\times \binom{5}{4} = 50$ ​​​​​​​