In a certain examination, the candidates can offer papers in English or Hindi or both the subjects. The number of candidates who appeared in the examination is `1000` of whom `650` appeared in English and `200` both in English and Hindi. `(i)` Hindi , `(ii)` English only, `(iii)` Hindi only

Answer

To solve the problem step by step, we will use the principles of set theory. Let's define the following:

- Let \( N \) be the total number of candidates who appeared for the examination.
- Let \( N_E \) be the number of candidates who appeared in English.
- Let \( N_H \) be the number of candidates who appeared in Hindi.
- Let \( N_{E \cap H} \) be the number of candidates who appeared in both English and Hindi.

Given data:
- \( N = 1000 \)
- \( N_E = 650 \)
- \( N_{E \cap H} = 200 \)

We need to find:
1. The number of candidates who opted for Hindi, \( N_H \).
2. The number of candidates who opted for English only.
3. The number of candidates who opted for Hindi only.

### Step 1: Find the total number of candidates who opted for Hindi

Using the principle of inclusion-exclusion for sets, we have:

\[ N = N_H + N_E - N_{E \cap H} \]

Rearranging this gives us:

\[ N_H = N + N_{E \cap H} - N_E \]

Substituting the known values:

\[ N_H = 1000 + 200 - 650 \]

Calculating:

\[ N_H = 1000 - 450 = 150 \]

So, the number of candidates who opted for Hindi is \( 150 \).

### Step 2: Find the number of candidates who opted for English only

The number of candidates who opted for English only can be calculated as:

\[ N_{E \text{ only}} = N_E - N_{E \cap H} \]

Substituting the known values:

\[ N_{E \text{ only}} = 650 - 200 \]

Calculating:

\[ N_{E \text{ only}} = 450 \]

So, the number of candidates who opted for English only is \( 450 \).

### Step 3: Find the number of candidates who opted for Hindi only

The number of candidates who opted for Hindi only can be calculated as:

\[ N_{H \text{ only}} = N_H - N_{E \cap H} \]

Substituting the known values:

\[ N_{H \text{ only}} = 150 - 200 \]

Calculating:

\[ N_{H \text{ only}} = 150 \]

So, the number of candidates who opted for Hindi only is \( 150 \).

### Summary of Results

1. Number of candidates who opted for Hindi: **150**
2. Number of candidates who opted for English only: **450**
3. Number of candidates who opted for Hindi only: **150**