Question

Bob is studying the effect of coral and sponge species on reef ecosystems using experiments in artificial square tanks. In each tank, he places 3 species of corals and 2 species of sponges. If there are 6 species of corals and 5 species of sponges to choose from, the minimum number of tanks required to test all combinations of 3 coral and 2 sponge species is ______ (Answer in integer)

Hint:

Count the ways to choose 3 corals out of 6 and the ways to choose 2 sponges out of 5 separately using combinations, then multiply the two counts together.

Answer 1

Correct Answer: 200

Step 1: Split the problem into two independent picks.
Setting up one tank means making two separate decisions: which 3 of the 6 corals go in, and which 2 of the 5 sponges go in. Neither decision affects the other, so we can count them apart and then combine.

Step 2: Count the coral choices by cancelling factors directly.
Picking 3 corals out of 6 without caring about order is \[ \binom{6}{3}=\frac{6\times5\times4}{3\times2\times1} \] The top multiplies to $120$ and the bottom multiplies to $6$, so this is $120/6=20$ possible coral groups.

Step 3: Count the sponge choices the same way.
Picking 2 sponges out of 5 without caring about order is \[ \binom{5}{2}=\frac{5\times4}{2\times1}=\frac{20}{2}=10 \] So there are $10$ possible sponge groups.

Step 4: Multiply, since every coral group can go with every sponge group.
Think of it as a grid: $20$ rows, one for each coral group, and $10$ columns, one for each sponge group. Every cell in that grid is a distinct tank setup that needs its own tank, so the count of cells is \[ 20\times10=200 \]

Step 5: Conclude.
Bob needs one tank for every cell in that grid to cover every combination once. \[ \boxed{200} \]