Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving combinations. Five cards marked respectively with the numbers \(1,2,3,4,\) and 5 are shuffled, and 2 cards are then drawn. How many different 2 -card hands are possible?

Step-by-Step Solution

Answer

There are 10 different 2-card hands possible.

Step 1: Understanding the Problem

We need to find the number of ways to choose 2 cards from a set of 5 cards numbered 1 to 5. This involves combinations because the order of selection does not matter.

Step 2: Using the Combination Formula

The combination formula is used to find the number of ways to choose a subset of items from a larger set, where order does not matter. It is given by:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \(n\) is the total number of items to choose from, \(k\) is the number of items to choose, and \(!\) denotes factorial.

Step 3: Substituting Values into the Formula

In our scenario, we have \(n = 5\) (since there are 5 cards) and \(k = 2\) (since we are choosing 2 cards). Substitute these values into the combination formula:\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} \]

Step 4: Simplifying the Expression

Cancel out the \(3!\) factors in the numerator and denominator:\[ C(5, 2) = \frac{5 \times 4}{2 \times 1} \]Calculate the product and division:\[ = \frac{20}{2} = 10 \]

Step 5: Conclusion

There are 10 different possible 2-card combinations that can be drawn from the set of 5 cards.

Key Concepts

Permutations

Permutations deal with the arrangement of objects in a specific order. Unlike combinations, the order here matters a lot. For example, the arrangement of the numbers 1, 2, and 3 as 123 is different from 321. The formula used to calculate permutations is:

• When all items are used: \[ P(n) = n! \]
• When selecting a subset of items: \[ P(n, k) = \frac{n!}{(n-k)!} \]

Here, \( n \) represents the total number of items and \( k \) is the number of items chosen. So, if you were asked to arrange 2 cards out of the 5 in order, permutations would be the right tool, considering cases like drawing \'2\' first, then \'3\' is different from drawing \'3\' first, then \'2\'. This contrasts with combinations, where such ordering is ignored.

Factorials

Factorials are a mathematical operation that multiplies a number by all the positive integers below it. Denoted by the symbol \(!\), factorials are a key part of calculating both permutations and combinations. For instance:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 0! = 1 \) by definition

This operation helps simplify complex calculations by breaking them down into manageable steps. In our exercise, we used factorials to compute combinations, which required cancelling out common terms to simplify the product.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It encompasses concepts such as permutations and combinations, which help in organizing and selecting items in different scenarios. In the exercise, combinatorics specifically focuses on combinations—choosing items without regard to order. Understanding combinatorics helps in solving various problems, like:

• Determining the number of different hand shapes possible in a card game
• Finding how many different lottery tickets can be created from a set of numbers

Overall, by understanding these principles, one can tackle more complex scenarios where distinguishing between order and selection becomes essential.

Probability

Probability is the measure of the likelihood that an event will occur. It's a core part of statistics and is used extensively in fields like finance, epidemiology, and engineering. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula is:\[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]In the context of our exercise, if you were to randomly draw two cards from the set of five, using our calculated combinations, the probability of any specific 2-card combination (such as drawing \'1\' and \'3\') would be computed as 1 out of the total number, which is 10 possible combinations. Probability differs from combinations in its focus on the chance of events happening rather than just the count of possibilities.