The number of distinct terms in the expansion of `(1+ 3x + 3x^(2) + x^(3))^(7)` is

Options

A. 18
B. 19
C. 28
D. 22

Answer

To find the number of distinct terms in the expansion of \( (1 + 3x + 3x^2 + x^3)^7 \), we can simplify the expression first.

### Step 1: Rewrite the expression
Notice that \( 1 + 3x + 3x^2 + x^3 \) can be rewritten as \( (1 + x)^3 \) because:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x)^3 \]
This is derived from the binomial expansion of \( (a + b)^3 \).

### Step 2: Raise to the power of 7
Now, we raise this expression to the power of 7:
\[ (1 + 3x + 3x^2 + x^3)^7 = ((1 + x)^3)^7 = (1 + x)^{21} \]

### Step 3: Determine the number of distinct terms
The expansion of \( (1 + x)^n \) contains \( n + 1 \) distinct terms. Therefore, for \( n = 21 \):
\[ \text{Number of distinct terms} = 21 + 1 = 22 \]

### Conclusion
Thus, the number of distinct terms in the expansion of \( (1 + 3x + 3x^2 + x^3)^7 \) is \( \boxed{22} \).

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