Explanation #1: Some say in the Gemara that Beis Shamai's teaching, that it is Hekdesh, refers to the white one. We can say that he intended to be Makdish what will leave his house first;

He said black, because he erred. He thought that a black ox will leave first.

Therefore, his mistake does not prevent Hekdesh from taking effect on the first ox that will leave his house. Hekdesh takes effect even on the white one.

Explanation #2: Some say that the first black ox that will leave after a white one is Kadosh, even though it did not leave first from the house;

The case is, it left the house first among the other black oxen. I will explain this in the Gemara.

Observation: Similarly, we can explain the cases of a Dinar and a barrel in two ways. Just like we explain the Reisha, we can explain these similarly.

The Gemara will explain why we need these clauses of a Dinar and a barrel of wine. (The Tosfosim that pertain to here are on Daf 33a, beginning with DH Mai Taima.)

Explanation: If one vowed to be a Nazir, and transgressed and drank wine during his Nezirus, and he asked Chachamim to permit his vow, even according to Beis Shamai, who say that She'elah does not apply to Hekdesh, and She'elah does not apply to Nezirus, like it says above (9a)...

Even so, according to Beis Shamai, he must accept Nezirus in a good, esteemed expression proper for acceptance of Nezirus;

However, there are many expressions or Kinuyim that are not acceptance. Even Beis Shamai agree that if one accepted in an inferior expression, it is not a vow.

Now, he asked a Chacham if the expression he accepted on himself is an expression of Nezirus, or not. Chachamim forbade him, to say that it is an expression of acceptance.

He transgressed his Nezirus initially. He drank wine. He counts from the time he vowed, and all the days he transgressed count towards his days of Nezirus;

We do not fine him to return to count like the days he counted while he was transgressing his Nezirus.

Inference: You are forced to say that we discuss when he transgressed. If not, obviously he counts from when he vowed!

Rather, you are forced to say that the simple meaning of our Mishnah, which teaches that he counts from the time he vowed, [teaches that] the days in which he transgressed count towards the days of his Nezirus;

Due to the Seifa, which teaches "and they permitted him", the Reisha teaches "they forbade", even though they did not forbid anything. Rather, they told him that he accepted Nezirus in a good expression.

Explanation: [The Chacham] told him that his acceptance had no effect. He did not vow in a good expression to accept Nezirus. It turns out that Nezirus did not take effect on him even for a moment.

If he had an animal separated, that he separated for his Nezirus before he asked Chachamim, it goes to graze in the herd with his animals, which are totally Chulin;

Since Nezirus did not take effect on him, there is no mistaken Hekdesh at all. Even Beis Shamai agree;

Only when he said "the black ox that will leave first will be Hekdesh", and a white ox left first, the white one is Kodesh, according to the opinion that it is because he does not care [about the color], and wants Hekdesh to take effect even on the white [ox];

He said black, because he erred. He thought that a black ox will leave first.

However, here that Nezirus did not take effect on him initially at all, and the animal was separated only for the need of Nezirus, and there is no Nezirus here, also there is no Hekdesh;

This is worse than mistaken Hekdesh. Therefore, even Beis Shamai agree that it goes to graze in the herd, since the vow was totally uprooted;

Since he said "for the Korbanos of my Nezirus", and he is not a Nazir, his words have no effect. He is like one who is not obligated, and he thinks that he is obligated Chatas, and he said "this animal is a Chatas...." His words have no effect, even according to Beis Shamai.

Inference: [Since you agree that it goes to graze in the herd,] this shows that mistaken Hekdesh is not Hekdesh. Just like you agree here, agree to us also about a black ox, about which you argue with us. You say that it is Hekdesh!

Implied question: Beis Shamai could have answered that here is worse than mistaken Hekdesh, like I explained!

Answer: They were not concerned to answer the real distinction. They answered them according to their (Beis Hillel's) words, how they understand their argument, and answered them from Ma'aser Behemah, which takes effect mistakenly, and it is Hekdesh.

Question: Why is a woman different than a woman whose husband annulled her? It says above (24a) that if the animal [that she separated for her Korban] was hers, a Chatas must die. Here, it goes to graze in the herd!

Answer #1: A husband cuts off.

Answer #2: Even if [a husband] uproots, in any case we established [the Mishnah] above to be like R. Elazar Hakapar. However, according to Rabanan it becomes Chulin;

Answer #3: Even according to R. Elazar, it seems that these are different. There, the vow took effect for a moment, for he annuls without regret or a Pesach [reason to consider it a mistake. Even though it is annulled retroactively, it was not a total mistake from the beginning]. Here, the vow did not take effect even for a moment, like I explained.

Explanation: It takes effect [to make the ninth and 11th Kadosh]. We should learn from it that mistaken Hekdesh takes effect!