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The Shulchan Aruch 199:10 writes that a child can be included for a zimun of three or 10(only one minor is counted),the Rama does not allow a minor to join.

1) If there is one adult Sephardi and one adult Ashkenazi and a Sephardi minor who ate together can the Ashkenazi count the minor?

2)If there are 8 Sephardim and one Ashkenazi and one sephardi minor,does majority rule against the Ashkenazi(diff from case one where its is one against one).

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1) In this first scenario, since Ashkenazim cannot include a child for a zimun, then clearly even in this case he cannot use a child. But DinOnLine.org here brings a different scenario, and says:

If an Ashkenazi child eats with two Sephardi men, the child can join the zimun and answer together with the men (Vezos Ha-Berachah p. 127).

2)With regard to the second scenario the same website writes:

If three Ashkenazi men join with six Sephardi men to form a zimun and a child is added to complete the minyan, the Sephardim should recite the zimun with the Name of Hashem, but the Ashkenazim should refrain from answering with the Name (Vezos Ha-Berachah p. 127).

Hence, as long as you have the majority of ten, i.e. six or more, who are Sephardi they can include a child in order to recite the zimun with Hashem's name.

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