Problem 40 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(34,2)\)

Step-by-Step Solution

Answer

The value of \( P(34,2) \) is 1122.

Step 1: Understanding the Permutation Notation

The notation \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. It is calculated using the formula:\[ P(n, r) = \frac{n!}{(n-r)!} \] where \( n! \) is the factorial of \( n \), which means \( n \times (n-1) \times (n-2) \times \, ... \, \times 2 \times 1 \).

Step 2: Plug in the Values

In our problem, we need to find \( P(34, 2) \). So, we substitute \( n = 34 \) and \( r = 2 \) into the permutation formula:\[ P(34, 2) = \frac{34!}{(34-2)!} = \frac{34!}{32!} \]

Step 3: Simplify the Factorial Expression

To simplify \( \frac{34!}{32!} \), notice that the 34 factorial will start as \( 34 \times 33 \times 32! \), which means the 32 factorial terms will cancel out:\[ \frac{34 \times 33 \times 32!}{32!} = 34 \times 33 \]

Step 4: Compute the Final Result

Calculate the product: \( 34 \times 33 \). Carrying out the multiplication, we have:\[ 34 \times 33 = 1122 \]

Step 5: Conclusion

Therefore, the number of permutations of 34 items taken 2 at a time is \( 1122 \).

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics and mathematics at large. You may encounter factorials often when dealing with permutations, combinations, and probabilities. The factorial of a whole number, represented by the symbol "!", is simply the product of all positive integers up to that number. For example, 5 factorial, written as 5!, is calculated as 5 × 4 × 3 × 2 × 1 = 120.

Factorials grow very rapidly in size; for instance, 10! is equal to 3,628,800, which is quite large! This rapid growth illustrates the vast number of ways you can arrange or order items.

If you see a factorial in an equation or problem, like in permutations, its role is vital in determining the total number of arrangements possible for a given set of items.

Permutation Formula

The permutation formula is a handy tool used to determine how many different ways you can arrange a subset of items from a given set. When using permutations, the order of the items is important, unlike in combinations where order does not matter.

The formula is as follows:

• \[ P(n, r) = \frac{n!}{(n-r)!} \]

Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to be arranged.

In our earlier example, we wanted to find \( P(34, 2) \). This means you are selecting 2 items out of a total of 34, and the order in which you pick those 2 items matters. By plugging the values into the formula, we find the number of permutations is 1,122.

Permutations vs Combinations

Understanding the difference between permutations and combinations is crucial when calculating possible arrangements or selections.

Permutations deal with arrangements where order matters. For example, if you are asked to arrange two colors out of a packet of red, blue, and green pencils, the order you pick the colors will lead to different arrangements. Red then blue is different from blue then red, which highlights the importance of order in permutations.

In contrast, combinations focus on groups where the order does not matter. If you choose two colors from the packet without worrying about the order, red and blue are the same as blue and red, effectively reducing the number of unique groups.

To sum up:

• Permutations: Order matters.
• Combinations: Order doesn't matter.

Once you know whether your scenario is best described by permutations or combinations, you can choose the right formula and find the number of possible arrangements or groups.