Question

What is the number of distinct terms in the expansion of $(x + y + z )25$ ?

• A. 25
• B. 50
• C. 125
• D. 325
• E. 351

Answer 1

Answer: (E)

To solve the problem of finding the number of distinct terms in the expansion of \((x + y + z)^{25}\), we use the formula for the number of distinct terms in the multinomial expansion.  

The general formula to determine the number of distinct terms in the expansion of \((x_1 + x_2 + ... + x_k)^n\) is given by the number of non-negative integer solutions to the equation:

\(a_1 + a_2 + ... + a_k = n\)

where \(a_i\) are the powers of each term \(x_i\) in the expansion. This is a classic "stars and bars" problem in combinatorics. The number of solutions is given by:

\(\binom{n + k - 1}{k - 1}\)

Here, \(n = 25\) and \(k = 3\) (since there are three variables x, y, and z). Substituting these values into the formula, we have:

\(\binom{25 + 3 - 1}{3 - 1} = \binom{27}{2}\)

Now, calculate \(\binom{27}{2}\):

\(\binom{27}{2} = \frac{27 \times 26}{2 \times 1} = \frac{702}{2} = 351\)

Thus, the number of distinct terms in the expansion of \((x + y + z)^{25}\) is 351.

Therefore, the correct answer is 351.