Yes Odessa, a broker can sell his own house--just like you can sell your own home. The difference is, a broker has more marketing muscle and can do a really good job in the marketing effort, probably better than you can, and so a broker may sell his house quicker than you would be able to.
As my brethern realtors have been clear to tell you, a broker must disclose he is a broker and it's his house, or if he has an interest in any house that he has for sale. He is also required to tell the world if he (or she) is an investor in the property that is for sale, maybe the realtor and a builder have made a partnership to build and market homes, that too needs to be disclosed to the world when the properties are advertised.
The law basically says the public may be too gullible or even too stupid to be able to deal with a realtor who has a house for sale, unless the realtor tells them it's his house; therefore realtors must disclose their ownership positions in any and all homes they are offering.