**Given Information:**

- Total number of boys surveyed = 100
- Number of boys using white shirts = 50
- Number of boys using red shirts = 40
- Number of boys using blue shirts = 30
- Number of boys using both white and red shirts = 20
- Number of boys using both red and blue shirts = 15
- Number of boys using both blue and white shirts = 10

**To Find:**

The number of boys using all the three colors (white, red, and blue).

**Solution:**

To find the number of boys using all three colors, we need to find the intersection of the sets of boys using white, red, and blue shirts.

Let's break down the given information using set notation:

- Let W represent the set of boys using white shirts.
- Let R represent the set of boys using red shirts.
- Let B represent the set of boys using blue shirts.

From the given information, we have:

- |W| = 50 (Number of boys using white shirts)
- |R| = 40 (Number of boys using red shirts)
- |B| = 30 (Number of boys using blue shirts)
- |W ∩ R| = 20 (Number of boys using both white and red shirts)
- |R ∩ B| = 15 (Number of boys using both red and blue shirts)
- |B ∩ W| = 10 (Number of boys using both blue and white shirts)

To find the number of boys using all three colors, we can use the formula:

|W ∩ R ∩ B| = |W| + |R| + |B| - |W ∩ R| - |R ∩ B| - |B ∩ W| + |W ∩ R ∩ B|

Substituting the given values:

|W ∩ R ∩ B| = 50 + 40 + 30 - 20 - 15 - 10 + |W ∩ R ∩ B|

Simplifying:

|W ∩ R ∩ B| = 95 + |W ∩ R ∩ B|

Now, we know that the total number of boys surveyed is 100. So, the number of boys using all three colors must be less than or equal to 100.

Therefore, we can conclude that the number of boys using all three colors is 5 (100 - 95).

Hence, there are 5 boys using all the colors (white, red, and blue).