Permutations - Algebra 2

2 men can be selected:

2 women can be selected out of 4 women:

Finally, after the selection process, these men and women can fill the executive body in ways.

This gives us a total of

This is a permutation of 10 objects (runners) taken 3 at a time, with no replacements.

Another way to look at this would be there are 10 runners competing for 1st place, 9 runners competing for 2nd place, and 8 runners competing for 3rd place.

In this problem, order is important because once someone is chosen as a position they can not be chosen again, and once a position is filled, no one else can fill that in mind.

The presidential spot has a possibility of 15 choices, then 14 choices for vice president and 13 for the treasurer.

So:

There are 7 letters in the word jubilee, so initially we can calculate that there are ways to re-arrange those letters. However, The letter e appears twice, so we're double counting. Divide by 2 factorial (2) to get .

At first, it makes sense that there are ways to re-arrange these letters. However, the letter A appears 3 times and the letter N appears twice, so divide first by 3 factorial and then 2 factorial:

To solve, evaluate

There are 7 potential actors and 2 different roles to fill. This would be calculated as divided by , or 42

There are 3 winners out of the total set of 13. That means we're calculating

Since order matters, use the permutation formula:

There are 5 letters to choose from (n), and you pick 4 of them (r).

So there are 120 possible words that can be formed.

Because the order is important, this is a permutation. There are 11 players to fill 3 spots, so .

Write the formula for permutation.

Evaluate .

Cancel all the common terms in the numerator and denominator.

The answer is:

Write the permutation formula.

Substitute the values in the given problem.

Cancel out the terms in the numerator and denominator.

The remaining terms in the numerator are:

The answer:

Write the permutation formula.

Substitute the values of the variables.

Simplify the factorials.

Cancel the common terms.

The answer is:

Write the permutation formula.

Substitute the values of into the formula.

Expand the terms.

Cancel out the terms in the denominator and denominator.

The remaining terms are:

Multiply these numbers.

The answer is:

Write the formula for permutation.

Substitute the values and expand the factorials.

Reduce the terms.

The answer is:

The formula to determine the permutation is:

Notice that for each number, there are 6 different choices since we have numbers ranging from zero to five, not five.

For a three digit lock, .

Substitute the terms into the formula.

The answer is:

Each of the two coin flips represents one event. The probability of obtaining exactly one heads and exactly one tails can be modeled as follows:

Event 1 * Event 2

The key to understanding this problem is to recognize that either heads followed by tails, or tails followed by heads, would satisfy the specific overall outcome asked for in the problem. Since Event 1 will produce either a heads or a tails, we have a 100% chance of obtaining an outcome from Event 1 that will satisfy one of the two requirements of the specific overall outcome (one heads and one tails). Event 2 will then have a 50% chance of producing an outcome that is the opposite, rather than the same, as the outcome of Event 1. We can therefore calculate the probability of our specific overall outcome as follows:

100% * 50%

1 * 0.5

0.5 = 50%

Therefore, the probability of obtaining one heads and one tails from two coin tosses is 50%.

There are five ways to choose the first scoop, then four ways to choose the second scoop, and finally three ways to choose the third scoop:

5 * 4 * 3 = 60