People with trisomy 21 develop Down’s syndrome. What law of Mendelian inheritance is violated in this disease? What is the most likely way this occurs?

Law of Segregation is violated.

Non-disjunction of chromosome pair 21 during anaphase I in gamete mother cell causes Down's syndrome.

Explanation:

Law of Segregation states that each gamete carries only one allele for each gene since the homologous chromosomes are segregated during gamete formation. When the homologous chromosomes from the pair 21 do not separate from each other during anaphase I, the some of the resultant gamete (sperm or egg) will have two copies of chromosome 21.

Fertilization between a gamete with two copies of chromosome 21 and the other normal gamete would result in a zygote with three copies of chromosome 21, that is Down's syndrome.