A woman has one X chromosome from her mother, and one from her father. Which of those two X chromosomes is passed down to her offspring? Is it always the one from the mother? Or always the one from the father? Or is it random? Please cite an online reference. Thanks!
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Random. Within Meiosis, sex chromosomes are treated no differently.
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With regards to your X-inactivation question, the X chromosome that gets inactivated is random as well. But say one cell has the maternal X chromosome inactivated, then the descendants of that cell will also have the maternal X chromosome inactivated. Say another cell has the paternal X chromosome inactivated, then the descendants of that cell will also have the paternal X chromosome inactivated. And all femals cells use only 1 X chromosome - that's how you get dosage compensation for females having 2 X chromosomes and males having 1 X chromosome.