Let T be the set of integers ( 3, 11, 19,27 .....451, 459,467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is

Options

A. 32
B. 28
C. 29
D. 30

Answer

To solve the problem, we need to determine the maximum possible number of elements in the subset \( S \) of the set \( T \) such that the sum of no two elements in \( S \) equals 470.

### Step 1: Identify the set \( T \)
The set \( T \) consists of integers starting from 3 and increasing by 8 each time, up to 467. We can express the elements of \( T \) as:
\[ T = \{3, 11, 19, 27, \ldots, 467\} \]

### Step 2: Determine the number of elements in \( T \)
The first term \( a = 3 \) and the common difference \( d = 8 \). The \( n \)-th term of an arithmetic progression can be given by the formula:
\[ a_n = a + (n - 1)d \]
Setting \( a_n = 467 \):
\[ 467 = 3 + (n - 1) \cdot 8 \]
Subtracting 3 from both sides:
\[ 464 = (n - 1) \cdot 8 \]
Dividing by 8:
\[ n - 1 = 58 \]
Thus,
\[ n = 59 \]
So, there are 59 elements in the set \( T \).

### Step 3: Identify pairs that sum to 470
We need to find pairs of elements in \( T \) whose sum equals 470. The pairs can be formed as follows:
- The first element \( x_1 = 3 \) pairs with \( x_2 = 467 \) (since \( 3 + 467 = 470 \))
- The second element \( x_3 = 11 \) pairs with \( x_4 = 459 \) (since \( 11 + 459 = 470 \))
- Continuing this way, we find pairs:
- \( (3, 467) \)
- \( (11, 459) \)
- \( (19, 451) \)
- \( (27, 443) \)
- ...

Continuing this process, we find that the pairs are:
- \( (3, 467) \)
- \( (11, 459) \)
- \( (19, 451) \)
- \( (27, 443) \)
- \( (35, 435) \)
- \( (43, 427) \)
- \( (51, 419) \)
- \( (59, 411) \)
- \( (67, 403) \)
- \( (75, 395) \)
- \( (83, 387) \)
- \( (91, 379) \)
- \( (99, 371) \)
- \( (107, 363) \)
- \( (115, 355) \)
- \( (123, 347) \)
- \( (131, 339) \)
- \( (139, 331) \)
- \( (147, 323) \)
- \( (155, 315) \)
- \( (163, 307) \)
- \( (171, 299) \)
- \( (179, 291) \)
- \( (187, 283) \)
- \( (195, 275) \)
- \( (203, 267) \)
- \( (211, 259) \)
- \( (219, 251) \)
- \( (227, 243) \)
- \( (235, 235) \) (this is the middle element)

### Step 4: Count the pairs
From the above pairs, we can see that there are 29 pairs, and since each pair can contribute only one element to the subset \( S \), we can choose one element from each pair.

### Step 5: Include the middle element
The middle element \( 235 \) does not have a pair that sums to 470, so we can include it in \( S \).

### Conclusion
Thus, the maximum number of elements in \( S \) is:
\[ 29 \text{ (from pairs)} + 1 \text{ (middle element)} = 30 \]

Therefore, the maximum possible number of elements in \( S \) is **30**.