Question

Flamingos Fanny and Freddy have three offspring: Happy, Glee, and Joy. These five flamingos are to be distributed to seven different zoos so that no zoo gets both a parent and a child :(. It is not required that every zoo gets a flamingo. In how many different ways can this be done? here if i am using case method on children i am getting a different answer than when i am using case methon on parents, why is that?

Answer 1

Solution $A$ :-

Case $1$ : Both parents are together

#ways $= 7 * 6^3$

Case $2$ : No parents together

#ways $= 7 * 6 * 5^3$

Final answer $= 7 * 6^3 + 7*6*5^3 = 6762$

Solution $B$ :-

Case $1$ : All children together

#ways $= 7* 6^2$

Case $2$ : Two children together

#ways $= {3 \choose 2} * 7 * 6 * 5^2$

Case $3$ : No children together

#ways $= 7*6*5*4^2$

Final answer $= 7*6^2 + {3 \choose 2} * 7 * 6 * 5^2 + 7*6*5*4^2 = 6762$