📘 Concept Explanation

When expanding products like \((x^{a}-m)(x^{b}-m)(x^{c}-p)\cdots\), the exponents that appear are the subset-sums of the chosen exponents \(\{a,b,c,\ldots\}\) (including the empty sum \(0\) from taking all constants). The count of distinct powers equals the number of distinct subset-sums produced by that exponent set.

For the three common patterns below, you can directly read off the total terms:

🧮 Formula(s)

1. First \(n\) natural powers \((1,2,\dots,n)\):
   Total terms \(=\;S_n+1=\dfrac{n(n+1)}{2}+1\).
2. First \(n\) even powers \((2,4,\dots,2n)\):
   Treat \(y=x^2\Rightarrow (1,2,\dots,n)\) in \(y\). Total terms \(=\;S_n+1=\dfrac{n(n+1)}{2}+1\).
   Exponents in \(x\) are all even, but the count of distinct exponents is unchanged.
3. First \(n\) odd powers \((1,3,5,\dots,2n-1)\):
   Total terms \(=\;n^2-1\) for \(n\ge 3\).
   Quick checks: \(n=3\Rightarrow 8\) terms; \(n=4\Rightarrow 15\) terms; \(n=5\Rightarrow 24\) terms. (For small \(n\): \(n=1\Rightarrow 2\) terms, \(n=2\Rightarrow 4\) terms.)

❓ Example Questions

1. Find the total number of terms in \((x-1)(x^{2}-2)(x^{3}-3)(x^{4}-4)\).
2. Find the total number of terms in \((x^{2}-1)(x^{4}-2)(x^{6}-2)(x^{8}-4)\).
3. Find the total number of terms in \((x-1)(x^{3}-2)(x^{5}-2)(x^{7}-4)\).

✅ Solution / Shortcut

1. Natural powers \(1,2,3,4\) \((n=4)\): \(\;S_4+1=\dfrac{4\cdot5}{2}+1=10+1=11\).
2. Even powers \(2,4,6,8\) \((n=4)\): let \(y=x^2\Rightarrow (1,2,3,4)\) in \(y\), so again \(S_4+1=11\).
3. Odd powers \(1,3,5,7\) \((n=4)\): use \(n^2-1=16-1=15\).

📝 Quick Quiz

How many distinct terms appear in the expansion of \((x-1)(x^{3}-2)(x^{5}-3)(x^{7}-4)(x^{9}-5)\)?