Parah, Chapter 9, Mishnah 7. This chapter deals with the various mishaps that can befall the ashes of the parah adumah and the chattas water made from them. Here the mishnah discusses ashes of the red cow, valid for sprinkling, that got mixed together with ordinary ashes: ashes of burnt wood or the like.
The mishnah's words are these. It opens with "Eifer kasher shenit'arev", valid ash that became blended, and continues "be'eifer mikleh", with ash of burnt material, ordinary ash that has no sanctity. The ruling then follows: "holchin achar harov letamei", meaning that in deciding whether the blend transmits impurity we look to whichever ash forms the larger portion. Yet "ve'ein mekadshin bo", the blend may not be used to sanctify water. Against this, "Rebbi Eliezer omer", Rebbi Eliezer holds otherwise: "mekadshin bekulan", one may perform the sanctification using the entire blend.
Following the majority for tumah
Regarding the ability of this mixture to convey tumah, the deciding factor is which component makes up the larger part. If the greater quantity is red cow ashes, the mixture transmits impurity both to one who touches it and to one who carries it. If the greater quantity is ordinary ashes, the mixture conveys no impurity at all.
But not for sanctifying water
For the purpose of sanctifying water, however, the mixture is unusable, and this holds true even where the red cow ashes are in the majority. Here the chachamim applied a stringency in the laws of the chattas: the principle that a minority is nullified in a majority does not operate. The ordinary ashes remain fully present in the mixture, and therefore the mixture cannot serve for kiddush.
Rebbi Eliezer reasons differently. Since genuine red cow ashes are definitely present somewhere in this mixture, in his view the entire mixture may be used to sanctify water. The halachah, however, does not follow Rebbi Eliezer.
Ashes that convey tumah
Notice what the mishnah takes for granted: that the rule of conveying tumah through touching and through carrying, which we know applies to sanctified chattas water, applies to the ashes of the red cow as well, and not only to the chattas water and the chattas ashes in their sprinkling form.
The Rambam's formulation, however, suggests that he read our mishnah differently: not as discussing the blended ash in its own right, but as discussing water that had already been sanctified with that blended ash. The Mishnah Acharonah understands it the same way, and rules that impurity follows whichever ash is present in greater quantity. On that reading, when the parah adumah ash forms the larger portion and water was sanctified with the blend, that water transmits tumah precisely as chattas water does. Ash that has not been used for kiddush, on the other hand, transmits nothing, since transmitting tumah means that a person is rendered impure until nightfall, and plain ash carries no impurity whatsoever.