jmc

Consider the polynomial $$\begin{array}{rl}p(x)& =\frac{a^4(x-b)(x-c)(x-d)(x-e)}{(a-b)(a-c)(a-d)(a-e)}+\frac{b^4(x-a)(x-c)(x-d)(x-e)}{(b-a)(b-c)(b-d)(b-e)}\\ & +\frac{c^4(x-a)(x-b)(x-d)(x-e)}{(c-a)(c-b)(c-d)(c-e)}+\frac{d^4(x-a)(x-b)(x-c)(x-e)}{(d-a)(d-b)(d-c)(d-e)}\\ & +\frac{e^4(x-a)(x-b)(x-c)(x-d)}{(e-a)(e-b)(e-c)(e-d)}.\end{array}$$Note that $p(x)$ is a polynomial of degree at most 4. Also, $p(a)=a^4,$ $p(b)=b^4,$ $p(c)=c^4,$ $p(d)=d^4,$ and $p(e)=e^4.$ Since the polynomial $p(x)$ and $x^4$ agree at five different values, by the Identity Theorem, they are the same polynomial.

The expression given in the problem is the coefficient of $x^4$ in $p(x),$ which is then $\boxed{1}.$