The television network MTV airs a show called “Are You The One?” Unlike many oth

Question

The television network MTV airs a show called “Are You The One?”

Unlike many other TV shows on the network, the show features a surprising

amount of mathematics. The premise of the show is that ten men and ten

women live in a house together. Secretly, the group has been split into ten

“correct” male-female pairs which a matchmaker believes to be especially

compatible. In other words, each player has exactly one “perfect match” of

the opposite sex, though they have no idea which other player is their perfect

match.

1. A

matchup ceremony

happens at the end of each episode, in which the

group of twenty players partition themselves into ten male-female pairs. How

many distinct matchup ceremonies are possible? If the group picks their pairs

randomly, what is the probability that all ten pairs at a single matchup cer-

emony are correct?

Explanation / Answer

The first female can be match with 10 males. The second female can be matched with the remaining 9 males. The third female can be matched with the remaining 8 males and so on. So, there can be

10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 10! distinct matchup ceremonies possible

The probability that all ten pairs at a single matchup ceremony are correct = (1 / 10!)

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