In a group of `1000` people, there are `750` people, who can speak Hindi and `400` people, who can speak Bengali.
`(i)` Find number of people who can speak Hindi only.
`(ii)` Number of people who can speak Bengali only is
`(iii)` Number of people who can speak both Hindi and Bengali is

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion.

### Given:
- Total number of people, \( N = 1000 \)
- Number of people who can speak Hindi, \( N(H) = 750 \)
- Number of people who can speak Bengali, \( N(B) = 400 \)

### Step 1: Find the number of people who can speak both Hindi and Bengali.

Using the principle of inclusion-exclusion, we can find the number of people who speak both languages as follows:

\[ N(H \cup B) = N(H) + N(B) - N(H \cap B) \]

Since \( N(H \cup B) \) cannot exceed the total number of people, we can rearrange the equation to find \( N(H \cap B) \):

\[ N(H \cap B) = N(H) + N(B) - N \]

Substituting the values:

\[ N(H \cap B) = 750 + 400 - 1000 \]

\[ N(H \cap B) = 150 \]

### Step 2: Find the number of people who can speak Hindi only.

To find the number of people who speak Hindi only, we subtract the number of people who speak both languages from the total number of Hindi speakers:

\[ N(H \text{ only}) = N(H) - N(H \cap B) \]

Substituting the values:

\[ N(H \text{ only}) = 750 - 150 \]

\[ N(H \text{ only}) = 600 \]

### Step 3: Find the number of people who can speak Bengali only.

Similarly, to find the number of people who speak Bengali only, we subtract the number of people who speak both languages from the total number of Bengali speakers:

\[ N(B \text{ only}) = N(B) - N(H \cap B) \]

Substituting the values:

\[ N(B \text{ only}) = 400 - 150 \]

\[ N(B \text{ only}) = 250 \]

### Final Answers:
(i) Number of people who can speak Hindi only: **600**
(ii) Number of people who can speak Bengali only: **250**
(iii) Number of people who can speak both Hindi and Bengali: **150**

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