The number of dissimilar terms in the expansion of `(a+2b+3c)^8` is

Options

A. 9
B. 24
C. 45
D. 10

Answer

To find the number of dissimilar terms in the expansion of \((a + 2b + 3c)^8\), we can use the multinomial theorem. The formula to calculate the number of distinct terms in the expansion of \((x_1 + x_2 + ... + x_k)^n\) is given by:

\[ \text{Number of terms} = \binom{n + k - 1}{k - 1} \]

where \(n\) is the power and \(k\) is the number of different variables.

### Step 1: Identify \(n\) and \(k\)

In our case:
- The expression is \((a + 2b + 3c)^8\).
- Here, \(n = 8\) (the power) and \(k = 3\) (the number of different terms: \(a\), \(2b\), and \(3c\)).

### Step 2: Apply the formula

Now, we can substitute the values of \(n\) and \(k\) into the formula:

\[ \text{Number of terms} = \binom{8 + 3 - 1}{3 - 1} = \binom{10}{2} \]

### Step 3: Calculate \(\binom{10}{2}\)

To calculate \(\binom{10}{2}\):

\[ \binom{10}{2} = \frac{10!}{2!(10 - 2)!} = \frac{10!}{2! \cdot 8!} \]

This simplifies to:

\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45 \]

### Conclusion

Thus, the number of dissimilar terms in the expansion of \((a + 2b + 3c)^8\) is **45**.

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