Interesting question. Here's how I think about it. If scores come in 10 point increments, and the scores range from 600 - 780, there is a total of 19 scores that students could have received. (600, 610, 620...780) If there are 19 possible scores one could have attained, the only way we can guarantee that no two people of the same gender/birth month received the same score would be if we have 19 or fewer people of each gender in each month. (In other words, if I have, say, 20 men in some month, I know for a fact that I'll have at least one duplicate score, as there are only 19 total possible scores for me to distribute to those 20 men.)

So let's say I have exactly 19 men and 19 women who are born in each month. That means I have 38 students born in each month, and a total of 38*12 = 456 students. If I have 478 students, that means I will have to have at least one month in which more than 19 men or more than 19 women were born. If there are more than 19 people of a given gender born in a given month, I know for a fact that I will have at least one duplicate score.