Problem 10 · Polynomials with p(1) = 1
$p(1)$ is the coefficient sum: counting signs that balance out.
Integer answer, at most 4 digitsConsider all polynomials $p(x)$ of degree $5$ whose coefficients are all $1$, $-1$ or $0$. How many of them satisfy $p(1) = 1$?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: $p(1)$ is the sum of the six coefficients, and the $x^5$ coefficient cannot be $0$ (degree $5$).
If the leading coefficient is $1$, the other five must sum to $0$: equally many $1$s and $-1$s. Choosing positions: $1 + 20 + 30 = 51$ ways ($k = 0, 1, 2$ pairs).
If the leader is $-1$, the other five must sum to $2$: two $1$s and no $-1$ ($\binom{5}{2} = 10$) or three $1$s and one $-1$ ($\binom{5}{3}\binom{2}{1} = 20$): $30$ ways.
$$51 + 30 = 81.$$
Answer: 81