A certain jar of jelly beans contains 50 black jelly beans, 50 red jelly beans, 50 green jelly beans, and...

1. Translate the problem requirements: We need to find the minimum number of jelly beans to remove that guarantees at least 14 beans of the same color, regardless of which beans we actually pick
2. Identify the worst-case scenario: To delay getting 14 of the same color as long as possible, we should remove as many beans as possible while keeping each color under 14
3. Calculate the maximum beans before guarantee: Determine how many beans we can remove while still avoiding 14 of any single color
4. Apply the guarantee principle: Add one more bean to force the guarantee, since the next bean must create a group of 14

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is really asking. We have a jar with 150 jelly beans total: 50 black, 50 red, and 50 green. We need to find the minimum number of jelly beans we must remove to guarantee that at least 14 beans have the same color.

The key word here is "guarantee" - this means we need to find the worst-case scenario. No matter how unlucky we are with our picks, we still need to be certain that we'll have 14 beans of the same color.

Process Skill: TRANSLATE - Converting the guarantee requirement into a worst-case analysis

2. Identify the worst-case scenario

To understand the worst case, let's think about it this way: imagine we're trying to avoid getting 14 beans of the same color for as long as possible. What's the most beans we could pick while still avoiding this?

The worst-case scenario would be if we keep picking beans in a way that delays getting 14 of any single color. This means we want to pick as evenly as possible from each color, but stop just before we get 14 of any color.

Since we want to avoid getting 14 of any color, the maximum we can have of any single color is 13. So in the worst case, we could pick:

- 13 black beans
- 13 red beans
- 13 green beans

3. Calculate the maximum beans before guarantee

Now let's count how many beans we can remove while still avoiding 14 of any single color:

\(13 + 13 + 13 = 39\) beans

This means we can remove up to 39 beans and still potentially avoid having 14 beans of the same color (in the worst-case scenario where we pick exactly 13 of each color).

4. Apply the guarantee principle

Here's the crucial insight: if we've already removed 39 beans (13 of each color), what happens when we remove one more bean?

That 40th bean must be either black, red, or green. No matter which color it is, we'll now have 14 beans of that color:

- If it's black: we'll have \(13 + 1 = 14\) black beans
- If it's red: we'll have \(13 + 1 = 14\) red beans
- If it's green: we'll have \(13 + 1 = 14\) green beans

Therefore, removing 40 beans guarantees that we have at least 14 beans of the same color, no matter what.

Process Skill: INFER - Drawing the non-obvious conclusion that the 40th bean must create a group of 14

4. Final Answer

The minimum number of jelly beans that must be removed to ensure at least 14 have the same color is 40.

This matches answer choice (B) 40.

To verify: we showed that 39 beans can be removed without guaranteeing 14 of the same color (worst case: 13 of each), but 40 beans definitely guarantees it (the 40th bean must make one color reach 14).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "guarantee" vs "possibility"

Students often confuse what "ensure" or "guarantee" means in probability problems. They might think about what's possible (like getting lucky and picking 14 of the same color early) rather than what's guaranteed in the worst-case scenario. This leads them to calculate based on favorable outcomes instead of the pigeonhole principle approach needed here.

2. Focusing on favorable scenarios instead of worst-case

Many students will try to calculate how quickly they could get 14 of the same color (best case) rather than identifying the scenario that delays this outcome the longest (worst case). They might think "I could get 14 black beans in just 14 picks" without realizing the question asks for a guarantee regardless of luck.

3. Misinterpreting the constraint structure

Some students may not recognize this as a pigeonhole principle problem. They might attempt to use combinations, permutations, or basic probability calculations instead of understanding that we need to find when it becomes impossible to avoid having 14 beans of the same color.

Errors while executing the approach

1. Incorrect distribution in worst-case scenario

Students might incorrectly distribute the beans when trying to delay getting 14 of any color. For example, they might think they can take all 50 of one color before moving to the next, not realizing that the worst case involves taking as evenly as possible (13 of each color) to delay the inevitable.

2. Off-by-one errors in counting

Even when students understand the approach, they often make counting errors. They might calculate that taking 13 of each color gives 39 beans, but then forget that the very next bean (the 40th) must guarantee the condition, leading them to answer 39 instead of 40.

Errors while selecting the answer

1. Selecting the "almost guarantee" number

Students who correctly identify that 39 beans can be taken without guaranteeing 14 of the same color might mistakenly select this as their final answer. They calculate the maximum number that doesn't guarantee the condition (39) rather than the minimum number that does guarantee it (40).

2. Choosing an answer that exceeds necessary bounds

Some students, unsure of their calculation, might select a "safe" higher number like 50 (thinking we need to potentially exhaust one entire color) or even higher values, not recognizing that 40 is sufficient and therefore the minimum required.