The number of distinct terms in `(a + b+ c + d + e)^3` is

Options

A. 35
B. 38
C. 42
D. 45

Answer

To find the number of distinct terms in the expression \((a + b + c + d + e)^3\), we can use the formula derived from the multinomial expansion. Here's the step-by-step solution:

### Step 1: Identify the number of variables and the power
In the expression \((a + b + c + d + e)^3\), we have:
- Number of distinct variables (terms) \(r = 5\) (which are \(a, b, c, d, e\))
- The power \(n = 3\)

### Step 2: Apply the formula for the number of distinct terms
The formula to find the number of distinct terms in the expansion of \((x_1 + x_2 + ... + x_r)^n\) is given by:
\[ \text{Number of distinct terms} = \binom{n + r - 1}{r - 1} \]
Substituting the values of \(n\) and \(r\):
\[ \text{Number of distinct terms} = \binom{3 + 5 - 1}{5 - 1} = \binom{7}{4} \]

### Step 3: Calculate \(\binom{7}{4}\)
Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
We calculate \(\binom{7}{4}\):
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} \]
Calculating the factorials:
\[ = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

### Conclusion
Thus, the number of distinct terms in the expansion of \((a + b + c + d + e)^3\) is \(35\).