In a survey of population of 450 people, it is found that 205 can speak English, 210 can speak Hindi and 120 people can speak Tamil. If 100 people can speak both Hindi and English, 80 people can speak both English and Tamil, 35 people can speak Hindi and Tamil and 20 people can speak all the three languages, find the number of people who can speak English but not a Hindi or Tamil. Find also the number of people who can speak neither English nor Hindi nor Tamil.

Explanation

To solve the problem, we will use the principle of inclusion-exclusion to find the number of people who can speak only English and then determine how many people can speak neither English, Hindi, nor Tamil.

### Step 1: Define the Sets
Let:
- \( E \) = the set of people who can speak English
- \( H \) = the set of people who can speak Hindi
- \( T \) = the set of people who can speak Tamil

From the problem, we have the following information:
- Total population, \( n(U) = 450 \)
- \( n(E) = 205 \) (people who can speak English)
- \( n(H) = 210 \) (people who can speak Hindi)
- \( n(T) = 120 \) (people who can speak Tamil)
- \( n(H \cap E) = 100 \) (people who can speak both Hindi and English)
- \( n(E \cap T) = 80 \) (people who can speak both English and Tamil)
- \( n(H \cap T) = 35 \) (people who can speak both Hindi and Tamil)
- \( n(E \cap H \cap T) = 20 \) (people who can speak all three languages)

### Step 2: Calculate the Number of People Who Speak Only English
To find the number of people who speak only English, we can use the formula:

\[ n(E \text{ only}) = n(E) - n(H \cap E) - n(E \cap T) + n(E \cap H \cap T) \]

Substituting the values:

\[ n(E \text{ only}) = 205 - 100 - 80 + 20 \]
\[ n(E \text{ only}) = 205 - 180 \]
\[ n(E \text{ only}) = 25 \]

### Step 3: Calculate the Number of People Who Speak Neither Language
Using the principle of inclusion-exclusion, we can find the total number of people who speak at least one language:

\[ n(E \cup H \cup T) = n(E) + n(H) + n(T) - n(H \cap E) - n(E \cap T) - n(H \cap T) + n(E \cap H \cap T) \]

Substituting the values:

\[ n(E \cup H \cup T) = 205 + 210 + 120 - 100 - 80 - 35 + 20 \]
\[ n(E \cup H \cup T) = 205 + 210 + 120 - 100 - 80 - 35 + 20 \]
\[ n(E \cup H \cup T) = 535 - 215 \]
\[ n(E \cup H \cup T) = 320 \]

Now, to find the number of people who speak neither language:

\[ n(\text{neither}) = n(U) - n(E \cup H \cup T) \]
\[ n(\text{neither}) = 450 - 320 \]
\[ n(\text{neither}) = 130 \]

### Final Answers
1. The number of people who can speak English but not Hindi or Tamil is **25**.
2. The number of people who can speak neither English, Hindi, nor Tamil is **130**.