Given the following experiments, tell whether order is important or not. Then so...

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August 2022 1 0 Report

Given the following experiments, tell whether order is important or not. Then solve each problem.
1. Choosing 3 winners out of the 10 semifinalists in a certain beauty pageant for the titles Grand Winner, First Runner-up, and Second Runner-up.
2. Choosing 3 distinct flavors of ice cream from 10 flavors and placing them on an ice cream cone.
3. Choosing 15 out of 40 students to join in the school’s Mathematics club.
4. Selecting 5 out of 7 available fruits in making fruit salad.
5. Selecting 4 officers from parents who will be President, Vice-President, Secretary, and Treasurer.

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MrFahrenheit

Verified answer

PERMUTATION vs COMBINATION

In permutation, the objects are arranged in such way that their order is a requirement. It means that the rank of each object should be considered in their arrangement.

On the other hand, in combination, the objects are arranged regardless of their order. This means that it does not require the rank of each object in their arrangement.

Permutation

In the given items, let us inspect which of the following experiments need the definite order of objects.

Here are the examples of permutation based on the given experiments:

Choosing 3 winners out of the 10 semifinalists in a certain beauty pageant for the titles Grand Winner, First Runner-up, and Second Runner-up.
Selecting 4 officers from parents who will be President, Vice-President, Secretary, and Treasurer.

Combination

Since we already listed above the examples of permutation, we will be heading to the listing of examples of combination. The following experiments can be classified as a combination:

Choosing 3 distinct flavors of ice cream from 10 flavors and placing them on an ice cream cone.
Choosing 15 out of 40 students to join in the school’s Mathematics club.
Selecting 5 out of 7 available fruits in making fruit salad.

The Formulae for Solving Permutation and Combination

The formula for solving for the permutation of a given situation /set is expressed as: $_{n}P_{r} = \frac{n!}{(n-r)!}$

where:

nPr = the total number of permutations
n = the total number of objects
r = the number of objects taken at a time

On the other hand, the formula for solving combination of a given situation/set is expressed as: $_{n}C_{r} = \frac{n!}{r!(n-r)!}$

Factorial Symbol (!)

The factorial symbol is denoted by an exclamation point (!). When a certain whole number is followed by a factorial symbol, this means that that number will multiply itself to all other whole numbers below it, excluding 0 and 1 because when you multiply any number by 1, the answer is still that number. So, this means that the endpoint of the factorial is 2.

To further explain this concept, here is an example. The expression "10!" can be expanded to 10x9x8x7x6x5x4x3x2.

Calculations

By direct substitution, we can solve for the permutations of items #1 and #5.

Let us start solving for item #1.

The n is 10, because there are a total of 10 semifinalists who competed for the crown. On the other hand, the r is 3 since there are only 3 rankings in the said beauty contest.

Using the formula given above, we will have:

$_{10}P_{3} = \frac{10!}{(10-3)!}\\_{10}P_{3} = \frac{10!}{7!} \\_{10}P_{3} = \frac{10x9x8x7x6x5x4x3x2}{7x6x5x4x3x2} \\_{10}P_{3} = 10x9x8\\_{10}P_{3} = 720$

Hence, there are 720 ways of choosing 3 winners out of the 10 semifinalists in the said beauty contest.

Solving for item #5:

The n is unknown since it was not specified in the given while the r is 4 since there are 4 slots for officers.

Still using the same formula, we will have:

$>On the other hand, items #2 to #4 can be solved by using the combination formula. Let us start with item #2.The n is 10 because there are 10 different ice cream flavors while r is 3 because there should be 3 distinct flavors.<img src=$

Hence, there are 120 ways of choosing 3 distinct flavors of ice cream from 10 flavors.

Solving for item #3 with the same formula:

The n is 40 since there are 40 students while the r is 15 since there should only be 15 students to join the Math Club.

$_{40}C_{15} = \frac{40!}{15!(40-15)!}\\_{40}C_{15} = \frac{40!}{15!25!} \\_{40}C_{15} = \frac{40x39x38x...x3x2}{15x14x13x...x3x2x25x24x23x...x3x2} \\_{40}C_{15} = \frac{40x39x38x...x27x26}{15x14x13x...x3x2} \\_{40}C_{15} = 4.023x10^{10}$

Hence, there are 4.023x10^10 ways of choosing 15 out of 40 students to join the Math Club.

Solving for item #4 with the same formula:

The n is 7 because there are 7 available fruits while the r is 5 since we can only select 5 out of 7.

$_{7}C_{5} = \frac{7!}{5!(7-5)!}\\_{7}C_{5} = \frac{7!}{5!2!}\\_{7}C_{5} = \frac{7x6x5x4x3x2}{5x4x3x2x2} \\_{7}C_{5} = \frac{7x6}{2} \\_{7}C_{5} = 7x3\\_{7}C_{5} = 21$

Hence, there are 21 ways of selecting 5 out of 7 available fruits in making fruit salad.

Another type of permutation is circular permutation where the objects are arranged in a circular way especially encircling a table. Here is an example of circular permutation: brainly.ph/question/12438067

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