The Animal Evolution

Probability Level 5

The jungle economy was divided into three sectors: Hunting, Entertaining and Protecting, each of which can accommodate exactly 6 animals. The only animals in the jungle are 6 lions, 6 bears, and 6 tigers. All animals are distinguishable. The lions are forbidden from working in the hunting sector, the bears are forbidden from working in the entertaining sector, and the tigers are forbidden from working in the protecting sector. In how many ways can the economy be divided among these animals if all the animals have to be given only one job?

The answer is 15184.

2 solutions

Ashish Menon
Jul 30, 2016

We can fill the hunting sector with bears and tigers, the entertaining sector with tigers and lions and the protecting sector with lions and bears. Suppose there are $x$ bears in the hunting sector, then there are $6 - x$ tigers in the same sector. Now, the remaining $x$ tigers can only be accomodated in the entertaining sector. So, there are $6 - x$ lions in the entertaing sector. The remaining $x$ lions can now be accomodated in the protecting sector along with the $6 - x$ remaining bears.

So, we see that the only determing factor is the number of bears we firstly accomodate in the hunting sector.

So, the number of ways of choosing $x$ bears in the hunting sector, $x$ tigers in the entertaining sector and $x$ lions in the enetertaining sector is $\dbinom{6}{x}$ . So, the total mumber of ways of choosing = $\dbinom{6}{x} × \dbinom{6}{x} × \dbinom{6}{x} = {\dbinom{6}{x}}^3$ .

Now, the value of $x$ can be anything between 0 to 6.
So, the number of ways of choosing = $\displaystyle \sum_{n=0}^{6} {\dbinom{6}{x}}^3 = \color{#3D99F6}{\boxed{15184}}$ .

The issue I had here is one of misinterpretation - I assumed it made no difference which of the 6 lions was in which job

Patrick Ramsey - 4 years, 10 months ago

Hehe, thats one typical mistake most people make. Not a bad error, surely you will not make that mistake next time :)

Ashish Menon - 4 years, 10 months ago

Since the question did not make it clear, the assumption that the different types of animal were indistinguishable was reasonable; in this case, there are only $7$ possible arrangements, with $a$ tigers hunting and $6-a$ tigers entertaining, $6-a$ bears hunting and $a$ bears protecting, $a$ lions entertaining and $6-a$ lions protecting for any $0 \le a \le 6$ .

Whether objects are distinguishable or indistinguishable is a key point in combinatoric questions, and a well set question will make clear which is meant each time.

Mark Hennings - 4 years, 10 months ago

@Mark Hennings – @Ashish Siva I agree that you need to make it clear that the animals are distinguishable.

Calvin Lin Staff - 4 years, 10 months ago

@Calvin Lin – Done, I'll take care next time.

Ashish Menon - 4 years, 10 months ago

Terrell Bombb
Aug 2, 2016

scenario 1: 36 x 6 = 216 Hunting: 1 Bear and 5 Tigers = 6C1 x 6C5 = 36 Entertainment: 1 Tiger and 5 Lions = 6C5 = 6 for choosing 5 lions from 6 and since the Hunting sector took care of choosing the Tigers, the Entertainment sector takes what's left Protecting: 1 Lion and 5 bears: the remaining animals left not chosen

scenario2: 225 x 15 = 3375 Hunting: 2 Bears and 4 Tigers = 6C2 x 6C4 = 15 x 15 = 225 Entertainment: 2 Tigers and 4 Lions = 6C4 = 15 similar explanation from scenario 1 Protecting: 2 Lions and 4 Bears: the remaining animals left not chosen

scenario3: 400 x 20 = 8000 Hunting: 3 Bears and 3 Tigers = 6C3 x 6C3 = 20 x 20 = 400 Entertainment: 3 Tigers and 3 Lions = 6C3 = 20 similar explanation from scenario 1 Protecting: 3 Lions and 3 Bears: the remaining animals left not chosen

scenario4: 225 x 15 = 3375 Hunting: 4 Bears and 2 Tigers = 6C4 x 6C2 = 15 x 15 = 225 Entertainment: 4 Tigers and 2 Lions = 6C2 = 15 similar explanation from scenario 1 Protecting: 4 Lions and 2 Bears: the remaining animals left not chosen

scenario5: 36 x 6 = 216 Hunting: 5 Bears and 1 Tigers = 6C5 x 6C1 = 36 Entertainment: 5 Tigers and 1 Lions = 6C1 = 6 similar explanation from scenario 1 Protecting: 2 Lions and 4 Bears: the remaining animals left not chosen

scenario6: 1 possible way to arrange the animals this way Hunting: 6 bears and 0 Tigers ENtertainment: 6 Tigers and 0 Lions Protecting: 6 Lions and 0 Bears

scenario7: 1 possible way to arrange the animals this way Hunting: 0 bears and 6 Tigers ENtertainment: 0 Tigers and 6 Lions Protecting: 0 Lions and 6 Bears

225 + 3375 + 8000 + 3375 + 225 + 1 + 1 = 15184

Nice way! :) +1

Ashish Menon - 4 years, 10 months ago

thank you! it's a bit long though.

Terrell Bombb - 4 years, 10 months ago

The Animal Evolution

The answer is 15184.

2 solutions

0 pending reports