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)
Species Combinations: Minimum Tanks Calculation
This solution details the calculation for the minimum number of experimental tanks needed based on combinations of coral and sponge species for Bob's study.
Coral Sponge Species Problem Analysis
The core task is to determine the total number of unique combinations possible when selecting a specific number of coral species and sponge species from larger sets. This requires applying the combination formula.
- Identify the number of ways to choose 3 coral species from 6 available.
- Identify the number of ways to choose 2 sponge species from 5 available.
- Multiply these two results to find the total minimum number of tanks required for all unique combinations.
Combination Calculation Steps
We use the combination formula, $C(n, k)$, which calculates the number of ways to choose $k$ items from a set of $n$ items without regard to the order of selection. The formula is:
$C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$1. Coral Species Calculation
Calculate the number of ways to choose 3 coral species from 6:
$C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$There are 20 distinct combinations for the coral species.
2. Sponge Species Calculation
Calculate the number of ways to choose 2 sponge species from 5:
$C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10$There are 10 distinct combinations for the sponge species.
3. Total Minimum Tanks Calculation
Multiply the number of coral combinations by the number of sponge combinations to find the total minimum tanks needed:
$ \text{Minimum Tanks} = C(6, 3) \times C(5, 2) = 20 \times 10 = 200 $The minimum number of tanks required is 200.
Important Questions from Combinations
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books ?
Let us assume a person climbing the stairs can take one stair or two stairs at a time. How many ways can this person climb a flight of eight stairs?
There are 9 different rock lizard species, each with a unique colour. Lizards of 2 different species sit on a rock at any given time. The number of possible colour combinations of rock lizards seen together on a rock is _________
(Answer in integer)- There are nine species of Impatiens (balsams) found in laterite plateaus of the northern Western Ghats, each with a distinct colour. If a plateau has exactly 6 species, then the number of possible colour combinations in the plateau is __________. (Answer in integer)
- The number of different possible ways of forming five intramolecular disulfide bonds with ten cysteine residues of a protein is ________
