It is known that if $ (A, A^*) $ is a Leonard pair, then the linear transformations $ A $, $ A^* $ satisfy the Askey-Wilson relations $ A^2A^* − \betaAA^*A + A^*A^2 − gamma(AA^* + A^*A) − sigmaA^* = gamma^*A^2 + omegaA + etaI, A^*2A − \betaA^*AA^* + AA^*2 − gamma^*(A^*A + AA^*) − sigma^*A = gammaA^*2 + omegaA^* + eta^*I $, for some scalars $ \beta $, $ gamma $, $ gamma^* $, $ sigma $, $ sigma^* $, $ omega $, $ eta $, $ eta^* $. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations. Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality 九州大学21世紀COEプログラム「機能数理学の構築と展開」