Question

The number of terms in the expansion of \( (2x + 3y + 5z)^5 \) is

• A. 18
• B. 6
• C. 21
• D. 15

Hint:

The number of terms in the multinomial expansion of \( (a + b + c)^n \) is given by \( \binom{n + k - 1}{k - 1} \), where \( k \) is the number of variables.

Answer 1

The Correct Option is C

Step 1: Identify the multinomial expansion.
In the expansion of \( (2x + 3y + 5z)^5 \), the number of terms corresponds to the number of distinct combinations of powers of \( x, y, z \) in the expansion. Each term in the expansion is of the form \( (2x)^a (3y)^b (5z)^c \), where \( a + b + c = 5 \) and \( a, b, c \geq 0 \).
Step 2: Apply the stars and bars method.
The number of distinct non-negative integer solutions to the equation \( a + b + c = 5 \) is given by the formula for combinations with repetition: \[ \binom{5 + 3 - 1}{3 - 1} = \binom{7}{2} = 21. \]
Step 3: Conclusion.
Thus, the number of terms in the expansion is 21, which corresponds to option (C).