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 ?
- 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)
- Ten teams participate in a tournament. Every team plays each of the other teams twice. The total number of matches to be played is
- The number of ways in which a supervisor can choose four workers out of 10 equally competent workers is ________.
- In a box, there are 5 green balls and 10 blue balls. A person picks 6 ballsrandomly. The probability that the person has picked 4 green balls and 2 blueballs is
