Question

Pablo is allowed to choose 1 of 3 different fruit beverages and 2 of 4 different healthy grain bars for his afternoon snack. How many different combinations does he have from which to choose?

Short answer

Pablo has 18 different combinations to choose from.

Step-by-step solution

Understand the Problem

Identify the different choices Pablo has. Pablo can choose 1 of 3 different fruit beverages and 2 of 4 different healthy grain bars.

Calculate Beverage Choices

Pablo has 3 options for fruit beverages.

Calculate Grain Bar Combinations

To decide how many ways Pablo can choose 2 out of 4 grain bars, we need to use the combination formula: \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} \]. Calculating this, we get: \[ \binom{4}{2} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1) \times (2 \times 1)} = 6. \] Therefore, there are 6 combinations.

Multiply the Choices

Multiply the number of beverage choices by the number of combinations of grain bars to find the total number of combinations. This gives us: \[ 3 \times 6 = 18. \]

Key concepts

Combination Formula

The combination formula is essential when we need to select groups from a larger set. Mathematically, it's represented as \(\binom{n}{k}\). This reads as 'n choose k'. The formula is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Where:

• \text{n} = total number of items.
• \text{k} = number of items to choose.
• \text{!} = factorial, which means the product of all positive integers up to that number.

For instance, in our exercise, we needed to find out how many ways Pablo can choose 2 grain bars out of 4, hence \(\binom{4}{2}\). Plugging into the formula: \(\frac{4!}{2! \times (4-2)!} = 6\).
This means there are 6 ways to select 2 grain bars from 4.

Probability Calculations

Probability calculations help us understand the likelihood of multiple events happening. For Pablo's snack choice, the total number of possible combinations is derived through multiplication.
Here's why:

• Pablo has 3 choices of fruit beverages.
• For each fruit beverage, there are 6 different combinations of grain bars.

Multiplying these choices: \[ 3 \times 6 = 18 \]
This shows that there are 18 different snack combinations Pablo can choose from.

Choosing Groups

Choosing groups involves selecting a subset of items from a larger pool. This commonly appears in statistical problems. In Pablo's case, he needs to choose 2 bars out of 4, often expressed as '4 choose 2'. This was solved using the combination formula.
The steps to solve such problems include:

• Identifying the total number of items (n).
• Deciding how many items to choose (k).
• Applying the combination formula: \( \binom{n}{k} \).

Understanding these steps allows for easy calculation of different group selections.