If `S = {x:x` is a positive multiple of `3` less than `100}` and `P = {x : x` is a prime number less than `20`}. Then, `n(S) + n(P)` is equal to (a) 34 (b) 31 (c) 33 (d)41

Options

A. `34`
B. `31`
C. `33`
D. `41`

Explanation

To solve the problem, we need to find the number of elements in the sets \( S \) and \( P \) and then sum these two values.

### Step 1: Define the set \( S \)
The set \( S \) is defined as:
\[ S = \{ x : x \text{ is a positive multiple of } 3 \text{ less than } 100 \} \]

### Step 2: Identify the positive multiples of 3 less than 100
The positive multiples of 3 are:
\[ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99 \]

### Step 3: Count the elements in set \( S \)
To count the multiples of 3 less than 100, we can use the formula for the number of terms in an arithmetic sequence:
\[ n = \frac{T - a}{d} + 1 \]
where:
- \( T \) is the last term (99),
- \( a \) is the first term (3),
- \( d \) is the common difference (3).

Substituting the values:
\[ n = \frac{99 - 3}{3} + 1 \]
\[ n = \frac{96}{3} + 1 \]
\[ n = 32 + 1 \]
\[ n = 33 \]

Thus, \( n(S) = 33 \).

### Step 4: Define the set \( P \)
The set \( P \) is defined as:
\[ P = \{ x : x \text{ is a prime number less than } 20 \} \]

### Step 5: Identify the prime numbers less than 20
The prime numbers less than 20 are:
\[ 2, 3, 5, 7, 11, 13, 17, 19 \]

### Step 6: Count the elements in set \( P \)
Counting the prime numbers listed:
There are 8 prime numbers.

Thus, \( n(P) = 8 \).

### Step 7: Calculate \( n(S) + n(P) \)
Now we can find the total:
\[ n(S) + n(P) = 33 + 8 = 41 \]

### Final Answer
Thus, the answer is \( \boxed{41} \).

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