There are 976 persons working in office X and each person is using at least one of the three different electronic devices i.e. iPhone, iPad and laptop. The ratio of the number of persons using iPad and laptop is 11:10 and the number of persons using iPhone is 20% more than the number of persons using the laptop. The number of persons using three devices is 124 and the number of persons using the iPhone is 50 more than the number of persons using the iPad. Ratio of the number of persons using only iPhone and only iPad is 17:15 and the number of persons using only laptop is 106. The number of persons using only laptop and iPad is 80% of the number of persons using only iPad.

To solve this problem, we'll use a combination of algebraic equations and logical reasoning. We'll define variables for the number of people using each device and their combinations.

Let's define:
- \( A \): Number of people using an iPhone
- \( B \): Number of people using an iPad
- \( C \): Number of people using a laptop
- \( x \): Number of people using only iPhone
- \( y \): Number of people using only iPad
- \( z \): Number of people using only laptop
- \( xy \): Number of people using only iPhone and iPad
- \( yz \): Number of people using only iPad and laptop
- \( zx \): Number of people using only laptop and iPhone
- \( xyz \): Number of people using all three devices

We are given:

1. \( xyz = 124 \) (Number of people using all three devices)
2. Ratio of persons using iPad to laptop is \( 11:10 \). So, let the number of persons using iPad be \( 11k \) and those using a laptop be \( 10k \). Thus, \( B = 11k \) and \( C = 10k \).
3. Number of persons using iPhone is 20% more than those using a laptop. So, \( A = 1.2C \). Using \( C = 10k \), we have \( A = 1.2 \times 10k = 12k \).
4. Number of persons using iPhone is 50 more than those using iPad. So, \( A = B + 50 \). Substituting \( A = 12k \) and \( B = 11k \), we get \( 12k = 11k + 50 \). This simplifies to \( k = 50 \).
5. Ratio of people using only iPhone to only iPad is \( 17:15 \). Let the number of people using only iPhone be \( 17m \) and only iPad be \( 15m \). So, \( x = 17m \) and \( y = 15m \).
6. Number of persons using only laptop is \( 106 \). So, \( z = 106 \).
7. Number of persons using only laptop and iPad is 80% of the number of persons using only iPad. So, \( yz = 0.8y = 0.8 \times 15m = 12m \).

Using \( k = 50 \), we can find \( A \), \( B \), and \( C \):
- \( A = 12k = 12 \times 50 = 600 \)
- \( B = 11k = 11 \times 50 = 550 \)
- \( C = 10k = 10 \times 50 = 500 \)

Now, let's set up an equation for the total number of persons using the devices:

\[
x + y + z + xy + yz + zx + xyz = 976
\]

Substituting the known values:

\[
17m + 15m + 106 + xy + 12m + zx + 124 = 976
\]

Combining like terms:

\[
44m + xy + zx + 230 = 976
\]

Simplifying, we have:

\[
44m + xy + zx = 746
\]

Now, let's use the relationships we have:

1. From the total number of people using iPhone \( A \):
   \[
   A = x + xy + zx + xyz \Rightarrow 600 = 17m + xy + zx + 124
   \]
   \[
   476 = 17m + xy + zx
   \]

2. From the total number of people using iPad \( B \):
   \[
   B = y + xy + yz + xyz \Rightarrow 550 = 15m + xy + 12m + 124
   \]
   \[
   414 = 27m + xy
   \]

Using the equation \( 44m + xy + zx = 746 \) and substituting \( 27m + xy = 414 \):

\[
17m + zx = 332
\]

Now, solving for \( m \) from \( 476 = 17m + xy + zx \) and \( 17m + zx = 332 \):

\[
476 - 332 = xy \Rightarrow xy = 144
\]

\[
27m + 144 = 414 \Rightarrow 27m = 270 \Rightarrow m = 10
\]

Substituting \( m = 10 \) into the equations:

- \( x = 17m = 170 \)
- \( y = 15m = 150 \)
- \( yz = 12m = 120 \)
- \( xy = 144 \)
- \( zx = 332 - 170 = 162 \)

Thus, the number of persons using each combination is:

- Only iPhone: \( 170 \)
- Only iPad: \( 150 \)
- Only laptop: \( 106 \)
- Only iPhone and iPad: \( 144 \)
- Only iPad and laptop: \( 120 \)
- Only laptop and iPhone: \( 162 \)
- All three devices: \( 124 \)

All values add up correctly to \( 976 \).

### Summary

The number of people using each combination is as follows:

1. **Only iPhone**: 170
2. **Only iPad**: 150
3. **Only laptop**: 106
4. **Only iPhone and iPad**: 144
5. **Only iPad and laptop**: 120
6. **Only laptop and iPhone**: 162
7. **All three devices**: 124

Do you have any questions, or would you like further details?

### Relative Questions
1. How would the solution change if the number of persons using only the laptop was different?
2. Can you create a Venn diagram to represent these combinations?
3. What if the ratio of iPhone to laptop users was different?
4. How would the solution change if no one used all three devices?
5. Can we calculate the number of people using exactly two devices from the given data?

### Tip

When working with multiple overlapping sets, using Venn diagrams or setting up equations with clearly defined variables helps to systematically solve the problem and ensure no cases are missed.

Explore how to solve a complex problem involving set theory and algebra. Understand the distribution of iPhone, iPad, and laptop users among 976 individuals in office X, including those using combinations of these devices. Detailed step-by-step solution provided.

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