Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
1. Consider the polynomial expression \((ax + b)^{100} - (cx + d)^{100}\). We need to find the values of \(n\) for which there are nonzero real numbers \(a, b, c, d\) such that the polynomial has exactly \(n\) nonzero coefficients after expanding and collecting similar terms.

2. Expand both terms using the binomial theorem:
   \[
   (ax + b)^{100} = \sum_{i=0}^{100} \binom{100}{i} a^i b^{100-i} x^i
   \]
   \[
   (cx + d)^{100} = \sum_{i=0}^{100} \binom{100}{i} c^i d^{100-i} x^i
   \]

3. Subtract the two expansions:
   \[
   (ax + b)^{100} - (cx + d)^{100} = \sum_{i=0}^{100} \binom{100}{i} (a^i b^{100-i} - c^i d^{100-i}) x^i
   \]

4. The coefficient of \(x^i\) in the resulting polynomial is:
   \[
   \binom{100}{i} (a^i b^{100-i} - c^i d^{100-i})
   \]

5. For the coefficient of \(x^i\) to be zero, we need:
   \[
   a^i b^{100-i} = c^i d^{100-i}
   \]

6. To find the values of \(n\), we need to consider the cases where the coefficients are zero or nonzero.

7. If \(a = c\) and \(b \neq d\), then:
   \[
   a^i b^{100-i} - a^i d^{100-i} = a^i (b^{100-i} - d^{100-i})
   \]
   This implies that the coefficients are zero only if \(b^{100-i} = d^{100-i}\). This can happen for exactly one value of \(i\), giving \(n = 100\).

8. If \(a \neq c\) and \(b = d\), then:
   \[
   a^i b^{100-i} - c^i b^{100-i} = b^{100-i} (a^i - c^i)
   \]
   This implies that the coefficients are zero only if \(a^i = c^i\). This can happen for exactly one value of \(i\), giving \(n = 100\).

9. If \(a = c\) and \(b = d\), then:
   \[
   a^i b^{100-i} - a^i b^{100-i} = 0
   \]
   This implies that all coefficients are zero, giving \(n = 0\).

10. If \(\frac{a}{b} = \frac{c}{d}\), then:
    \[
    \left(\frac{a}{b}\right)^i = \left(\frac{c}{d}\right)^i
    \]
    This implies that either all coefficients are zero or none are zero, giving \(n = 0\) or \(n = 101\).

11. If \(\frac{a}{b} = -\frac{c}{d}\), then:
    \[
    \left(\frac{a}{b}\right)^i = -\left(\frac{c}{d}\right)^i
    \]
    This implies that the coefficients for odd \(i\) are zero, and the coefficients for even \(i\) are nonzero, giving \(n = 50\).

12. Therefore, the possible values of \(n\) are \(0, 50, 100, 101\).

The final answer is \(\boxed{0, 50, 100, 101}\)