Discussion
How many distinct terms are there in the expansion of \((w + x + y + z)^{10}\) ?
Answer: D
There are 286 distinct terms.
The terms of the expansion are the monomials \(w^{a} x^{b} y^{c} z^{d}\) with \(a + b + c + d = 10\) and each exponent a non-negative integer. Counting such solutions is the stars-and-bars problem: 10 stars and 3 bars in a row.
\[ \begin{aligned} \binom{10 + 4 - 1}{4 - 1} &= \binom{13}{3} \\ &= \frac{13 \times 12 \times 11}{6} = 286 \end{aligned} \]
Each such monomial appears with a positive multinomial coefficient, so none cancels. The figures 224, 228 and 281 do not come from the correct count; a common slip is \(\binom{12}{3} = 220\), which uses \(n + k - 2\) instead of \(n + k - 1\).
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