Let f(n) denotes the number of different ways, the positive integer n ca be expressed as the sum of the 1's and 2's. for example, f(4)=5.
i.e., `4=1+1+1+1`
`=1+1+2=1+2+1=2+1+1=2+2`
Q. The value of f{f(6)} is

Options

A. 376
B. 377
C. 321
D. 370

Explanation

To solve for \( f(f(6)) \), we need to first determine \( f(6) \) and then find \( f(13) \) since \( f(6) = 13 \).

### Step 1: Calculate \( f(1) \)
The number of ways to express 1 as a sum of 1's and 2's:
- Only one way: \( 1 \)

Thus, \( f(1) = 1 \).

### Step 2: Calculate \( f(2) \)
The number of ways to express 2:
- Two ways: \( 1 + 1 \) and \( 2 \)

Thus, \( f(2) = 2 \).

### Step 3: Calculate \( f(3) \)
The number of ways to express 3:
- Three ways: \( 1 + 1 + 1 \), \( 1 + 2 \), and \( 2 + 1 \)

Thus, \( f(3) = 3 \).

### Step 4: Calculate \( f(4) \)
The number of ways to express 4:
- Five ways: \( 1 + 1 + 1 + 1 \), \( 1 + 1 + 2 \), \( 1 + 2 + 1 \), \( 2 + 1 + 1 \), and \( 2 + 2 \)

Thus, \( f(4) = 5 \).

### Step 5: Calculate \( f(5) \)
The number of ways to express 5:
- Eight ways: \( 1 + 1 + 1 + 1 + 1 \), \( 1 + 1 + 1 + 2 \), \( 1 + 1 + 2 + 1 \), \( 1 + 2 + 1 + 1 \), \( 2 + 1 + 1 + 1 \), \( 2 + 2 + 1 \), \( 1 + 2 + 2 \), and \( 2 + 1 + 2 \)

Thus, \( f(5) = 8 \).

### Step 6: Calculate \( f(6) \)
The number of ways to express 6:
- \( f(6) = f(5) + f(4) = 8 + 5 = 13 \)

Thus, \( f(6) = 13 \).

### Step 7: Calculate \( f(7) \)
Using the recurrence relation:
- \( f(7) = f(6) + f(5) = 13 + 8 = 21 \)

Thus, \( f(7) = 21 \).

### Step 8: Calculate \( f(8) \)
- \( f(8) = f(7) + f(6) = 21 + 13 = 34 \)

Thus, \( f(8) = 34 \).

### Step 9: Calculate \( f(9) \)
- \( f(9) = f(8) + f(7) = 34 + 21 = 55 \)

Thus, \( f(9) = 55 \).

### Step 10: Calculate \( f(10) \)
- \( f(10) = f(9) + f(8) = 55 + 34 = 89 \)

Thus, \( f(10) = 89 \).

### Step 11: Calculate \( f(11) \)
- \( f(11) = f(10) + f(9) = 89 + 55 = 144 \)

Thus, \( f(11) = 144 \).

### Step 12: Calculate \( f(12) \)
- \( f(12) = f(11) + f(10) = 144 + 89 = 233 \)

Thus, \( f(12) = 233 \).

### Step 13: Calculate \( f(13) \)
- \( f(13) = f(12) + f(11) = 233 + 144 = 377 \)

Thus, \( f(13) = 377 \).

### Conclusion
The value of \( f(f(6)) = f(13) = 377 \).

### Final Answer
The value of \( f(f(6)) \) is **377**.