Sefirat HaOmer and Mourning Rebbi Akiva's Students: Two Which are Really One

There are two seemingly disparate aspects of the time period between Pesach and Shavuot: 1) Counting from our physical freedom to our spiritual freedom, when we received the Torah (אין לך בן חורין אלא מי שעוסק בתורה - Avot 6:2) and 2) mourning for Rebbi Akiva's 24,000 students. What is the connection between the two?

Why does the Gemara (Yevamot 62) describe his students as 12,000 pairs of students and not 24,000 students?

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Delivered at the OU Israel Center, May 23, 2019 (Lag BaOmer 5779)