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Denote by P"(x) the Legendre polynomial of degree n and let I_ I",..."k: P"(x)...P"k(x)dx. I",...,n is written as a sum involving binomial coefficients and the sum is interpreted via a combinatorial model. This makes possible a combinatorial proof of a number of general theorems concerning I",..."k, not all of which seem analytically straightforward, including a direct combinatorial derivation of the known formula for Ia,b, and the expression of Ia,b,c,d as a simple finite sum. In addition, adoi:10.1137/0519109 fatcat:adackwudanf63p6vka34vh2gta