Question:
The number of terms in the expansion of $(x + y + z)^{10}$ is:
The number of terms in the expansion of $(x + y + z)^{10}$ is:
Show Hint
For a trinomial $(x+y+z)^n$, the number of terms simplifies to $\frac{(n+1)(n+2)}{2}$. For $n=10$: $\frac{11 \times 12}{2} = 66$.
Updated On: Jun 3, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Concept
The number of terms in the multinomial expansion of $(x_1 + x_2 + \dots + x_r)^n$ is given by the formula $^{n+r-1}C_{r-1}$.
Step 2: Meaning
Here, we have $3$ variables ($x, y, z$), so $r = 3$, and the power is $n = 10$.
Step 3: Analysis
Substitute $n = 10$ and $r = 3$ into the formula: \[ \text{Number of terms} = ^{10+3-1}C_{3-1} = ^{12}C_2 \] Calculate $^{12}C_2$: \[ ^{12}C_2 = \frac{12 \times 11}{2 \times 1} = 6 \times 11 = 66 \]
Step 4: Conclusion
The expansion of $(x + y + z)^{10}$ contains exactly $66$ distinct terms.
Final Answer: (B)
The number of terms in the multinomial expansion of $(x_1 + x_2 + \dots + x_r)^n$ is given by the formula $^{n+r-1}C_{r-1}$.
Step 2: Meaning
Here, we have $3$ variables ($x, y, z$), so $r = 3$, and the power is $n = 10$.
Step 3: Analysis
Substitute $n = 10$ and $r = 3$ into the formula: \[ \text{Number of terms} = ^{10+3-1}C_{3-1} = ^{12}C_2 \] Calculate $^{12}C_2$: \[ ^{12}C_2 = \frac{12 \times 11}{2 \times 1} = 6 \times 11 = 66 \]
Step 4: Conclusion
The expansion of $(x + y + z)^{10}$ contains exactly $66$ distinct terms.
Final Answer: (B)
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