Before us is Mishnah 3 in the eleventh chapter of Yevamos, which deals with a case of children who became mixed up and its implications for the order of chalitzah and yibum.
The text of the Mishnah:
"Chamesh nashim" - five women who are not related to one another.
"Shenis'arvu veladoseihen" - their children became mixed up with one another while they were still infants, and it is not known which child belongs to which mother.
"Gadlu hata'arovos" - the group of mixed-up infants, whose mothers cannot be identified, grew up.
"Venas'u nashim umeisu" - they married women, and afterward they died.
Understanding the case:
Each of the mothers had an additional son whose lineage is known and clear. So we have two groups before us: five definite brothers, who know who their parents are, and five from the mixture, whose parents are unknown - and all of them died. The definite brothers do not know who is whose brother, and therefore each one of them is a possible yavam to each one of the widows.
The order of the chalitzos and the yibum:
"Arba'ah choltzin le'achas" - four of the five definite brothers perform chalitzah with one of the widows, since they do not know whether she is their yevamah.
"Ve'echad meyabmah" - the fifth brother, who did not perform chalitzah with her, performs yibum with her. She is not the woman with whom he did chalitzah, so he is permitted to do so, since perhaps she really is his yevamah.
"Ushloshah choltzin le'acheres" - the one who performed yibum, together with three of the others, performs chalitzah with a second widow - that makes four chalitzos.
"Ve'echad meihem" - and one of them, the one remaining who did not perform chalitzah with her, performs yibum with her. And the same procedure is followed with each and every one of the yevamos.
"Nimtze'u arba chalitzos veyibum lechol achas ve'achas" - in the end, four chalitzos and one yibum are performed for each one of the women.
Why was precisely this order chosen?
Seemingly a simpler arrangement could have been used: those same four brothers would perform chalitzah with each and every one of the women, and the fifth brother would perform yibum with all of them. From the standpoint of halachic permissibility, this approach would also have worked out well.
But the Gemara explains the advantage of the method in which they rotate in a circle, with each of the five brothers performing yibum with a different woman: in this way it is possible that every single yavam actually performs yibum with his true yevamah. If, on the other hand, one brother were to perform yibum with all of them, at most one of those yibumin would be with the correct yevamah, and all the rest certainly would not be.
In summary: In this Mishnah we learned the law of five women whose children became mixed up, where the children of the mixture married women and died: four chalitzos and one yibum are performed for each of the widows. We examined the rotation in which each of the definite brothers performs yibum with a different woman, and the reason for it - that in this way there is a possibility that every single yibum is performed between the correct yavam and his yevamah.