In the group of 85 people speak three language English, French and Japanese. 15 people speak only the French language, 10 people speak only the English language. 10 people speak both French and English language, 20 people speak both Japanese and French language and 15 people speak both English and Japanese language. 10 people speak all three kinds of language. Find how many people speak only Japanese language?
(a)15
(b)10
(c)5
(d)12

Options

A. 15
B. 10
C. 5
D. 12

Answer

To find the number of people who speak only Japanese, we can use the principle of inclusion-exclusion and the information given in the problem. Let's break down the steps:

### Step 1: Define the Variables
Let:
- \( E \): Number of people who speak English
- \( F \): Number of people who speak French
- \( J \): Number of people who speak Japanese

### Step 2: Gather the Given Information
From the problem, we have:
- \( |F \text{ only}| = 15 \) (only French)
- \( |E \text{ only}| = 10 \) (only English)
- \( |E \cap F| = 10 \) (both English and French)
- \( |F \cap J| = 20 \) (both French and Japanese)
- \( |E \cap J| = 15 \) (both English and Japanese)
- \( |E \cap F \cap J| = 10 \) (all three languages)

### Step 3: Calculate the Total Number of People
The total number of people is given as 85.

### Step 4: Calculate the Number of People Speaking Each Language
Using the information provided, we can calculate the total number of people who speak at least one language:

1. **Only French**: \( |F \text{ only}| = 15 \)
2. **Only English**: \( |E \text{ only}| = 10 \)
3. **French and English but not Japanese**: \( |E \cap F| - |E \cap F \cap J| = 10 - 10 = 0 \)
4. **French and Japanese but not English**: \( |F \cap J| - |E \cap F \cap J| = 20 - 10 = 10 \)
5. **English and Japanese but not French**: \( |E \cap J| - |E \cap F \cap J| = 15 - 10 = 5 \)
6. **All three languages**: \( |E \cap F \cap J| = 10 \)

### Step 5: Calculate the Total Count from the Above
Now, let's sum these values to find the total number of people who speak at least one language:

\[ \text{Total} = |F \text{ only}| + |E \text{ only}| + |E \cap F| + |F \cap J| + |E \cap J| + |E \cap F \cap J| \]
\[ = 15 + 10 + 0 + 10 + 5 + 10 = 50 \]

### Step 6: Find the Number of People Speaking Only Japanese
Now, we need to find the number of people who speak only Japanese. We know the total number of people is 85, so:

\[ \text{People speaking only Japanese} = 85 - \text{Total} \]
\[ = 85 - 50 = 35 \]

### Step 7: Calculate the Number of People Speaking Only Japanese
Now we need to account for those who speak Japanese but not only Japanese. We have:

- People who speak both Japanese and French (including those who speak all three): 20
- People who speak both Japanese and English (including those who speak all three): 15
- People who speak all three languages: 10

Thus, the number of people who speak Japanese but not only Japanese is:

\[ \text{Japanese speakers} = |F \cap J| + |E \cap J| - |E \cap F \cap J| = 20 + 15 - 10 = 25 \]

### Step 8: Calculate Only Japanese Speakers
Finally, we can find the number of people who speak only Japanese:

\[ \text{Only Japanese} = \text{Total Japanese speakers} - \text{Japanese speakers not only} = 35 - 25 = 10 \]

### Conclusion
The number of people who speak only Japanese is **10**.