In a group of 50 persons, 30 like tea, 25 like cofffee and 16 like both, how many like.
(i) either tea or coffee ?
(ii) neither tea nor coffee?

Explanation

To solve the problem step by step, we will use the principles of set theory.

### Given:
- Total number of persons (Universal Set, U) = 50
- Number of persons who like tea (T) = 30
- Number of persons who like coffee (C) = 25
- Number of persons who like both tea and coffee (T ∩ C) = 16

### (i) To find the number of persons who like either tea or coffee (T ∪ C):

1. **Use the formula for the union of two sets:**
\[ n(T \cup C) = n(T) + n(C) - n(T \cap C) \]
where:
- \( n(T \cup C) \) = number of persons who like either tea or coffee
- \( n(T) \) = number of persons who like tea
- \( n(C) \) = number of persons who like coffee
- \( n(T \cap C) \) = number of persons who like both tea and coffee

2. **Substitute the values into the formula:**
\[ n(T \cup C) = 30 + 25 - 16 \]

3. **Calculate the result:**
\[ n(T \cup C) = 55 - 16 = 39 \]

Thus, the number of persons who like either tea or coffee is **39**.

### (ii) To find the number of persons who like neither tea nor coffee:

1. **Use the formula for the number of persons who like neither tea nor coffee:**
\[ n(\text{Neither T nor C}) = n(U) - n(T \cup C) \]
where:
- \( n(U) \) = total number of persons
- \( n(T \cup C) \) = number of persons who like either tea or coffee (calculated in part (i))

2. **Substitute the values into the formula:**
\[ n(\text{Neither T nor C}) = 50 - 39 \]

3. **Calculate the result:**
\[ n(\text{Neither T nor C}) = 11 \]

Thus, the number of persons who like neither tea nor coffee is **11**.

### Summary of Results:
- (i) Number of persons who like either tea or coffee: **39**
- (ii) Number of persons who like neither tea nor coffee: **11**