Question

The number of terms in the expansion of $(x + y + z)^{10}$ is:

• A. 55
• B. 66
• C. 45
• D. 78

Hint:

For a trinomial $(x+y+z)^n$, the number of terms simplifies to $\frac{(n+1)(n+2)}{2}$. For $n=10$: $\frac{11 \times 12}{2} = 66$.

Answer 1

The Correct Option is B

Step 1: Counting terms in a multinomial.
The number of different terms in $(x_1 + \cdots + x_r)^n$ is $^{n+r-1}C_{r-1}$. This counts how many ways the power $n$ can be shared among the variables.

Step 2: Identify the numbers.
Here we have three letters $x, y, z$, so $r = 3$, and the power is $n = 10$.

Step 3: Plug into the formula.
\[ \text{terms} = {}^{10 + 3 - 1}C_{3-1} = {}^{12}C_2 \]

Step 4: Write $^{12}C_2$.
\[ ^{12}C_2 = \frac{12 \times 11}{2 \times 1} \]

Step 5: Simplify.
$12 \div 2 = 6$, so it becomes $6 \times 11$.

Step 6: Final count.
\[ 6 \times 11 = 66 \] \[ \boxed{ 66 \text{ terms} } \]