On Monday, a lab assistant was asked to prepare the samples for measurement for
each of the remaining days of the week (Tuesday to Friday). They were told that
20 samples must be chosen for measurement on Tuesday and 5 fewer samples
each next day until Friday.
How many ways are there to select the samples for the 4 days of measurement?
Show your work. 

To solve this problem, we need to determine how many ways we can distribute the samples for measurement from Tuesday to Friday, based on the conditions given. Let's break it down step by step.

### Step 1: Define the sample counts for each day
- On **Tuesday**, 20 samples must be chosen.
- On **Wednesday**, 5 fewer samples than Tuesday are required, i.e., 20 - 5 = 15 samples.
- On **Thursday**, again 5 fewer than Wednesday, i.e., 15 - 5 = 10 samples.
- On **Friday**, 5 fewer than Thursday, i.e., 10 - 5 = 5 samples.

Thus, the number of samples to be selected each day is:
- Tuesday: 20 samples
- Wednesday: 15 samples
- Thursday: 10 samples
- Friday: 5 samples

### Step 2: Total number of samples to be selected
The total number of samples to be selected from Tuesday to Friday is the sum of the daily samples:
\[
20 + 15 + 10 + 5 = 50
\]
Therefore, a total of 50 samples must be selected.

### Step 3: Total number of samples available
We do not have information about how many samples are available in total, but if we assume there is no restriction on the total number of available samples (i.e., the samples are distinct and there's no limitation on the pool), then we are essentially choosing **20 samples for Tuesday**, **15 samples for Wednesday**, **10 samples for Thursday**, and **5 samples for Friday** from a large pool.

### Step 4: Calculate the number of ways to select samples
Since the samples are distinct, the order in which we choose the samples matters (because they are being selected for different days). This is a classic combinatorial problem where we are choosing groups of samples on different days.

1. First, choose **20 samples for Tuesday** from the pool of \( n \) available samples. The number of ways to do this is \( \binom{n}{20} \).
2. After selecting the 20 samples for Tuesday, we now have \( n - 20 \) samples left in the pool. From these remaining samples, choose **15 samples for Wednesday**. The number of ways to do this is \( \binom{n-20}{15} \).
3. After selecting the 15 samples for Wednesday, we now have \( n - 35 \) samples left. From these, choose **10 samples for Thursday**, which gives \( \binom{n-35}{10} \).
4. Finally, after selecting the 10 samples for Thursday, we have \( n - 45 \) samples left, and from these, choose **5 samples for Friday**, which gives \( \binom{n-45}{5} \).

### Step 5: Combine the selections
The total number of ways to select the samples for Tuesday through Friday is the product of all these binomial coefficients:
\[
\binom{n}{20} \times \binom{n-20}{15} \times \binom{n-35}{10} \times \binom{n-45}{5}
\]

This represents the total number of ways to select the samples for each day, assuming that all \( n \) samples are distinct and the order matters.

---

Would you like to go over any particular step in more detail, or ask any related questions?

Here are 5 related questions to think about:
1. How would the problem change if the total number of available samples was limited?
2. What if the samples were not distinct? How would that affect the calculation?
3. Can you generalize this solution to a scenario where the sample reduction is not constant (i.e., not a fixed 5)?
4. How would you calculate the number of ways if there were restrictions on which samples could be selected for which day?
5. How would the problem change if the sample sizes for each day were random?

**Tip:** Understanding how to apply combinations and binomial coefficients to different selection scenarios is key in combinatorics, especially when order and restrictions are involved.

On Monday, a lab assistant was asked to prepare the samples for measurement for each of the remaining days of the week (Tuesday to Friday). They were told that 20 samples must be chosen for measurement on Tuesday and 5 fewer samples each next day until Friday. How many ways are there to select the samples for the 4 days of measurement? Show your work.

Discover how to solve a combinatorics problem involving the selection of samples over multiple days. Learn how to calculate the number of ways to select 20, 15, 10, and 5 samples from a pool, using binomial coefficients. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation