Question

The number of terms in the expansion of (x + y + z)¹⁰ is

• A. 66
• B. 142
• C. 11
• D. 110

Answer 1

The Correct Option is A

The number of terms in the expansion of \((x_1 + x_2 + ... + x_m)^n\) is given by \(^{n+m-1}C_{m-1}\) or \(^{n+m-1}C_n\).

In our case, \(n = 10\) and \(m = 3\) (since we have three terms: x, y, and z). 

So the number of terms is \(^{10+3-1}C_{3-1} = ^{12}C_2\).

\(^{12}C_2 = \frac{12!}{2!10!} = \frac{12 \times 11}{2 \times 1} = 6 \times 11 = 66\).

Answer: (A) 66

Answer 2

The number of terms in the expansion of $ (x_1 + x_2 + \dots + x_k)^n $ is given by:

$$ \binom{n+k-1}{k-1} \quad \text{or equivalently} \quad \binom{n+k-1}{n}. $$

Here, $ n = 10 $ and $ k = 3 $. Thus:

$$ \binom{10+3-1}{3-1} = \binom{12}{2} = \frac{12 \cdot 11}{2} = 66. $$

Final Answer: The final answer is $ {66} $.